unit 4 solving quadratic equations homework 10

Unit 4: Solving Quadratic Equations

Why are we studying this.

In this unit of study, students will continue to deepen their knowledge and understand of quadratic functions. Students will learn how to solve quadratic equations algebraically and interpret them in terms of the graph and in context.

Unit Schedule and Assignments

Weekly online, notebook: unit 4, table of contents & pages, hw #8: due 11/30, hw #10: due 12/14, hw #9: due 12/7, hw #11: due 12/21.

Daily Homework Assignments (w/Answer Keys)

Graphing: practice page.

Factoring Quadratic Expressions: Practice Page

8.1: #3-19 ODD

8.2: #1-15 odd, 9.1: #4-15 all + 18, 19, 22, 9.2: #1-15 odd + 23, 24, 9.3: #1-13 odd + 15-20 all, unit assessments, unit quizzes.

Mid-Unit Quiz #1: Solving Quadratics by Graphing and Factoring - Tuesday, December 4th (Per. 1 & 3) and Wednesday, December 5th (Per. 2)

Mid-Unit Quiz #2: Solving Quadratics by Square Roots, Completing the Square, and Quadratic Formula (POP QUIZ)

Unit 4 Assessment: Tuesday, December 18th (Per. 1 & 3) and Wednesday, December 19th (Per. 2)

Solving Quadratic Equations by Graphing

Unit Foldable: Solving Quadratic Equations

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Practice Page

Factoring Quadratic Expressions

In Class Slides (Day 1)

Factoring Trinomials Match Up

Factoring Quadratic Expressions (Orduna - video)

8.1-8.2: Solving Quadratic Equations by Factoring

8.1 Textbook Pages

8.2 Textbook Pages

In Class Slides (Day 2)

Notes/Practice Page

Solving Quadratic Equations by Factoring (Orduna - video)

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8.1: Math on the Spot Video

8.2: Math on the Spot Video

9.1: Solving Quadratic Equations by Square Roots

9.1 Textbook Pages

Notes/Practice Pages

Choose Your Own Adventure

9.2: Solving Quadratic Equations by Completing the Square

9.2 Textbook Pages

Coloring Page

9.3: Solving Quadratics by Quadratic Formula

9.3 Textbook Pages

Quadratic Formula Tic-Tac-Toe

UNIT REVIEW RESOURCES

Word Problem Practice

Solving by Different Methods

Quadratics Puzzle

4 to 1 Review Game

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Quadratic Equation Worksheets (pdfs)

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Enjoy these free sheets. Each one has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Plus each one comes with an answer key.

• Solve Quadratic Equations by Factoring
• Solve Quadratic Equations by Completing the Square
• Quadratic Formula Worksheet (real solutions)
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• Quadratic Formula Worksheet (both real and complex solutions)
• Discriminant Worksheet
• Sum and Product of Roots
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• NCERT Solutions
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• Chapter 4: Quadratic Equations

NCERT Solutions for Class 10 Maths Chapter 4 Quadratic Equations

Ncert solutions class 10 maths chapter 4 – cbse free pdf download.

NCERT Solutions Class 10 Maths Chapter 4 Quadratic Equations  contain all the solutions to the problems provided in the Class 10 Maths NCERT textbook for CBSE exam preparations. The questions from every section are framed and solved accurately by the subject experts. NCERT Solutions for Class 10 are detailed and step-by-step guides to all the queries of the students. The exercises present in the chapter should be dealt with utmost sincerity if one wants to score well in the examinations. Maths is a subject that requires a good understanding and a lot of practice. The tips and tricks to solve the problems easily are also provided here. A quadratic equation in the variable x is an equation of the form ax 2 + bx + c = 0, where a, b, c are real numbers, a ≠ 0. That is, ax 2 + bx + c = 0, a ≠ 0 is called the standard form of a quadratic equation.

Download Exclusively Curated Chapter Notes for Class 10 Maths Chapter – 4 Quadratic Equations

Download most important questions for class 10 maths chapter – 4 quadratic equations.

Quadratic equations arise in several situations around us. Hence, students should give special attention to learning the concepts related to this chapter of the latest CBSE Syllabus for 2023-24 thoroughly to excel in Class 10 Maths examinations. NCERT Solutions help the students in learning these concepts as well as in evaluating themselves. Practising these solutions repeatedly is bound to help the students in overcoming their shortcomings. Maths has either a correct answer or a wrong one. Therefore, it is imperative to concentrate while solving the questions to score full marks.

• Chapter 1 Real Numbers
• Chapter 2 Polynomials
• Chapter 3 Pair of Linear Equations in Two Variables
• Chapter 4 Quadratic Equations
• Chapter 5 Arithmetic Progressions
• Chapter 6 Triangles
• Chapter 7 Coordinate Geometry
• Chapter 8 Introduction to Trigonometry
• Chapter 9 Some Applications of Trigonometry
• Chapter 10 Circles
• Chapter 11 Constructions
• Chapter 12 Areas Related to Circles
• Chapter 13 Surface Areas and Volumes
• Chapter 14 Statistics
• Chapter 15 Probability
• Exercise 4.1
• Exercise 4.2
• Exercise 4.3
• Exercise 4.4

NCERT Solutions Class 10 Maths Chapter 4 – Quadratic Equations

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Access answers to NCERT Class 10 Maths Chapter 4 – Quadratic Equations

Exercise 4.1 page: 73.

1. Check whether the following are quadratic equations:

(i) (x + 1) 2 = 2(x – 3)

(ii) x 2 – 2x = (–2) (3 – x)

(iii) (x – 2)(x + 1) = (x – 1)(x + 3)

(iv) (x – 3)(2x +1) = x(x + 5)

(v) (2x – 1)(x – 3) = (x + 5)(x – 1)

(vi) x 2 + 3x + 1 = (x – 2) 2

(vii) (x + 2) 3 = 2x (x 2 – 1)

(viii) x 3 – 4x 2 – x + 1 = (x – 2) 3

(x + 1) 2 = 2(x – 3)

By using the formula for (a+b) 2 = a 2 +2ab+b 2

⇒ x 2  + 2x + 1 = 2x – 6

⇒ x 2  + 7 = 0

The above equation is in the form of ax 2 + bx + c = 0

Therefore, the given equation is a quadratic equation.

(ii) Given, x 2 – 2x = (–2) (3 – x)

⇒ x 2  –   2x = -6 + 2x

⇒ x 2  – 4x + 6 = 0

(iii) Given, (x – 2)(x + 1) = (x – 1)(x + 3)

By multiplication,

⇒ x 2  – x – 2 = x 2  + 2x – 3

⇒ 3x – 1 = 0

The above equation is not in the form of ax 2 + bx + c = 0

Therefore, the given equation is not a quadratic equation.

(iv) Given, (x – 3)(2x +1) = x(x + 5)

⇒ 2x 2  – 5x – 3 = x 2  + 5x

⇒  x 2  – 10x – 3 = 0

(v) Given, (2x – 1)(x – 3) = (x + 5)(x – 1)

⇒ 2x 2  – 7x + 3 = x 2  + 4x – 5

⇒ x 2  – 11x + 8 = 0

The above equation is in the form of ax 2  + bx + c = 0.

(vi) Given, x 2  + 3x + 1 = (x – 2) 2

By using the formula for (a-b) 2 =a 2 -2ab+b 2

⇒ x 2  + 3x + 1 = x 2  + 4 – 4x

⇒ 7x – 3 = 0

(vii) Given, (x + 2) 3  = 2x(x 2  – 1)

By using the formula for (a+b) 3  = a 3 +b 3 +3ab(a+b)

⇒ x 3  + 8 + x 2  + 12x = 2x 3  – 2x

⇒ x 3  + 14x – 6x 2  – 8 = 0

(viii) Given, x 3  – 4x 2  – x + 1 = (x – 2) 3

By using the formula for (a-b) 3  = a 3 -b 3 -3ab(a-b)

⇒  x 3  – 4x 2  – x + 1 = x 3  – 8 – 6x 2   + 12x

⇒ 2x 2  – 13x + 9 = 0

2. Represent the following situations in the form of quadratic equations:

(i) The area of a rectangular plot is 528 m 2 . The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.

(ii) The product of two consecutive positive integers is 306. We need to find the integers.

(iii) Rohan’s mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age.

(iv) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken

(i) Let us consider,

The breadth of the rectangular plot = x m

Thus, the length of the plot = (2x + 1) m

As we know,

Area of rectangle = length × breadth = 528 m 2

Putting the value of the length and breadth of the plot in the formula, we get,

(2x + 1) × x = 528

⇒ 2x 2  + x =528

⇒ 2x 2  + x – 528 = 0

Therefore, the length and breadth of the plot satisfy the quadratic equation, 2x 2  + x – 528 = 0, which is the required representation of the problem mathematically.

(ii) Let us consider,

The first integer number = x

Thus, the next consecutive positive integer will be = x + 1

Product of two consecutive integers = x × (x +1) = 306

⇒ x 2  + x = 306

⇒ x 2  + x – 306 = 0

Therefore, the two integers x and x+1 satisfy the quadratic equation, x 2  + x – 306 = 0, which is the required representation of the problem mathematically.

(iii) Let us consider,

Age of Rohan’s = x  years

Therefore, as per the given question,

Rohan’s mother’s age = x + 26

After 3 years,

Age of Rohan’s = x + 3

Age of Rohan’s mother will be = x + 26 + 3 = x + 29

The product of their ages after 3 years will be equal to 360, such that

(x + 3)(x + 29) = 360

⇒ x 2  + 29x + 3x + 87 = 360

⇒ x 2  + 32x + 87 – 360 = 0

⇒ x 2  + 32x – 273 = 0

Therefore, the age of Rohan and his mother satisfies the quadratic equation, x 2  + 32x – 273 = 0, which is the required representation of the problem mathematically.

(iv) Let us consider,

The speed of the train = x   km/h

Time taken to travel 480 km = 480/x km/hr

As per second condition, the speed of train = ( x  – 8) km/h

Also given, the train will take 3 hours to cover the same distance.

Therefore, time taken to travel 480 km = (480/x)+3 km/h

Speed × Time = Distance

( x  – 8)(480/ x  )+ 3 = 480

⇒ 480 + 3 x  – 3840/ x  – 24 = 480

⇒ 3 x  – 3840/ x  = 24

⇒ x 2  – 8 x  – 1280 = 0

Therefore, the speed of the train satisfies the quadratic equation, x 2  – 8 x  – 1280 = 0, which is the required representation of the problem mathematically.

Exercise 4.2 Page: 76

1. Find the roots of the following quadratic equations by factorisation:

(i) x 2  – 3x – 10 = 0 (ii) 2x 2  + x – 6 = 0 (iii) √2 x 2  + 7x + 5√2 = 0 (iv) 2x 2  – x +1/8 = 0 (v) 100x 2  – 20x + 1 = 0

(i) Given,  x 2  – 3 x  – 10 =0

Taking L.H.S.,

=> x 2  – 5 x  + 2 x  – 10

=> x ( x  – 5) + 2( x  – 5)

=>( x  – 5)( x  + 2)

The roots of this equation, x 2  – 3 x  – 10 = 0 are the values of x for which ( x  – 5)( x  + 2) = 0

Therefore,  x  – 5 = 0 or  x  + 2 = 0

=>  x  = 5 or  x  = -2

(ii) Given, 2 x 2  +  x  – 6 = 0

=> 2 x 2  + 4 x  – 3 x  – 6

=> 2 x ( x  + 2) – 3( x  + 2)

=> ( x  + 2)(2 x  – 3)

The roots of this equation, 2 x 2  +  x  – 6=0 are the values of x for which ( x x  + 2)(2 x  – 3) = 0

Therefore,  x  + 2 = 0 or 2 x  – 3 = 0

=>  x  = -2 or  x  = 3/2

(iii) √2  x 2  + 7 x  + 5√2=0

=> √2  x 2  + 5 x  + 2 x  + 5√2

=>  x  (√2 x  + 5) + √2(√2 x  + 5)= (√2 x  + 5)( x  + √2)

The roots of this equation, √2  x 2  + 7 x  + 5√2=0 are the values of x for which (√2 x  + 5)( x  + √2) = 0

Therefore, √2 x  + 5 = 0 or  x  + √2 = 0

=>  x  = -5/√2 or  x  = -√2

(iv) 2 x 2  –  x  +1/8 = 0

=1/8 (16 x 2   – 8 x  + 1)

= 1/8 (16 x 2   – 4 x  -4 x  + 1)

= 1/8 (4 x (4 x    – 1) -1(4 x  – 1))

= 1/8 (4 x  – 1) 2

The roots of this equation, 2 x 2  –  x  + 1/8 = 0, are the values of x for which (4 x  – 1) 2 = 0

Therefore, (4 x  – 1) = 0 or (4 x  – 1) = 0

⇒  x  = 1/4 or  x  = 1/4

(v) Given, 100x 2  – 20x + 1=0

= 100x 2  – 10x – 10x + 1

= 10x(10x – 1) -1(10x – 1)

= (10x – 1) 2

The roots of this equation, 100x 2  – 20x + 1=0, are the values of x for which (10x – 1) 2 = 0

∴ (10x – 1) = 0 or (10x – 1) = 0

⇒x = 1/10 or x = 1/10

2. Solve the problems given in Example 1.

Represent the following situations mathematically:

(i) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. We would like to find out how many marbles they had to start with.

(ii) A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day. On a particular day, the total cost of production was Rs. 750. We would like to find out the number of toys produced on that day.

(i) Let us say the number of marbles John has = x

Therefore, the number of marble Jivanti has = 45 – x

After losing 5 marbles each,

Number of marbles John has = x  – 5

Number of marble Jivanti has = 45 – x  – 5 = 40 –  x

Given that the product of their marbles is 124.

∴ ( x  – 5)(40 –  x ) = 124

⇒  x 2  – 45 x  + 324 = 0

⇒  x 2  – 36 x  – 9 x  + 324 = 0

⇒  x ( x  – 36) -9( x  – 36) = 0

⇒ ( x  – 36)( x  – 9) = 0

Thus, we can say,

x  – 36 = 0 or  x  – 9 = 0

⇒  x  = 36 or  x  = 9

If John’s marbles = 36

Then, Jivanti’s marbles = 45 – 36 = 9

And if John’s marbles = 9

Then, Jivanti’s marbles = 45 – 9 = 36

(ii) Let us say the number of toys produced in a day is x .

Therefore, cost of production of each toy = Rs(55 –  x )

Given the total cost of production of the toys = Rs 750

∴  x (55 –  x ) = 750

⇒  x 2  – 55 x  + 750 = 0

⇒  x 2  – 25 x  – 30 x  + 750 = 0

⇒  x ( x  – 25) -30( x  – 25) = 0

⇒ ( x  – 25)( x  – 30) = 0

Thus, either x  -25 = 0 or  x  – 30 = 0

⇒  x  = 25 or  x  = 30

Hence, the number of toys produced in a day will be either 25 or 30.

3. Find two numbers whose sum is 27 and product is 182.

Let us say the first number is x, and the second number is 27 – x.

Therefore, the product of two numbers

x(27 – x) = 182

⇒ x 2  – 27x – 182 = 0

⇒ x 2  – 13x – 14x + 182 = 0

⇒ x(x – 13) -14(x – 13) = 0

⇒ (x – 13)(x -14) = 0

Thus, either, x = -13 = 0 or x – 14 = 0

⇒ x = 13 or x = 14

Therefore, if first number = 13, then second number = 27 – 13 = 14

And if first number = 14, then second number = 27 – 14 = 13

Hence, the numbers are 13 and 14.

4. Find two consecutive positive integers, the sum of whose squares is 365.

Let us say the two consecutive positive integers are x  and  x  + 1.

Therefore, as per the given questions,

x 2  + ( x  + 1) 2  = 365

⇒  x 2  +  x 2  + 1 + 2 x  = 365

⇒ 2 x 2  + 2x – 364 = 0

⇒  x 2  +  x  – 182 = 0

⇒  x 2  + 14 x  – 13 x  – 182 = 0

⇒  x ( x  + 14) -13( x  + 14) = 0

⇒ ( x  + 14)( x  – 13) = 0

Thus, either, x  + 14 = 0 or  x  – 13 = 0,

⇒  x  = – 14 or  x  = 13

Since the integers are positive, x can be 13 only.

∴  x  + 1 = 13 + 1 = 14

Therefore, two consecutive positive integers will be 13 and 14.

5. The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.

Let us say the base of the right triangle is x cm.

Given, the altitude of right triangle = (x – 7) cm

From Pythagoras’ theorem, we know,

Base 2  + Altitude 2  = Hypotenuse 2

∴ x 2  + (x – 7) 2  = 13 2

⇒ x 2  + x 2  + 49 – 14x = 169

⇒ 2x 2  – 14x – 120 = 0

⇒ x 2  – 7x – 60 = 0

⇒ x 2  – 12x + 5x – 60 = 0

⇒ x(x – 12) + 5(x – 12) = 0

⇒ (x – 12)(x + 5) = 0

Thus, either x – 12 = 0 or x + 5 = 0,

⇒ x = 12 or x = – 5

Since sides cannot be negative, x can only be 12.

Therefore, the base of the given triangle is 12 cm, and the altitude of this triangle will be (12 – 7) cm = 5 cm.

6. A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was Rs.90, find the number of articles produced and the cost of each article.

Let us say the number of articles produced is x .

Therefore, cost of production of each article = Rs (2 x  + 3)

Given the total cost of production is Rs.90

∴  x (2 x  + 3) = 90

⇒ 2 x 2  + 3 x  – 90 = 0

⇒ 2 x 2  + 15 x  -12 x  – 90 = 0

⇒  x (2 x  + 15) -6(2 x  + 15) = 0

⇒ (2 x  + 15)( x  – 6) = 0

Thus, either 2 x  + 15 = 0 or  x  – 6 = 0

⇒  x  = -15/2 or  x  = 6

As the number of articles produced can only be a positive integer, x can only be 6.

Hence, the number of articles produced = 6

Cost of each article = 2 × 6 + 3 = Rs 15

Exercise 4.3 Page: 87

1. Find the roots of the following quadratic equations, if they exist, by the method of completing the square:

(i) 2 x 2  – 7 x  +3 = 0

(ii) 2 x 2  +  x  – 4 = 0 (iii) 4 x 2  + 4√3 x  + 3 = 0

(iv) 2 x 2  +  x  + 4 = 0

(i) 2 x 2  – 7 x  + 3 = 0

⇒ 2 x 2  – 7 x  = – 3

Dividing by 2 on both sides, we get

⇒ x 2 -7x/2 = -3/2

⇒ x 2 -2 × x ×7/4 = -3/2

On adding (7/4) 2 to both sides of the equation, we get

⇒ (x) 2 -2×x×7/4 +(7/4) 2 = (7/4) 2 -3/2

⇒ (x-7/4) 2 = (49/16) – (3/2)

⇒(x-7/4) 2 = 25/16

⇒(x-7/4) 2 = ±5/4

⇒  x  = 7/4 ± 5/4

⇒  x  = 7/4 + 5/4 or x = 7/4 – 5/4

⇒ x = 12/4 or x = 2/4

⇒ x = 3 or x = 1/2

(ii) 2x 2  + x – 4 = 0

⇒ 2x 2  + x = 4

Dividing both sides of the equation by 2, we get

⇒ x 2  +x/2 = 2

Now on adding (1/4) 2  to both sides of the equation, we get,

⇒ (x) 2  + 2 × x × 1/4 + (1/4) 2  = 2 + (1/4) 2

⇒ (x + 1/4) 2  = 33/16

⇒ x + 1/4 = ± √33/4

⇒ x = ± √33/4 – 1/4

⇒ x = (± √33-1)/4

Therefore, either x = (√33-1)/4 or x = (-√33-1)/4

(iii) 4x 2  + 4√3x + 3 = 0

Converting the equation into a 2 +2ab+b 2 form, we get,

⇒ (2x) 2  + 2 × 2x × √3 + (√3) 2  = 0

⇒ (2x + √3) 2  = 0

⇒ (2x + √3) = 0 and (2x + √3) = 0

Therefore, either x = -√3/2 or x = -√3/2

(iv) 2x 2  + x + 4 = 0

⇒ 2x 2  + x = -4

⇒ x 2  + 1/2x = 2

⇒ x 2  + 2 × x × 1/4 = -2

By adding (1/4) 2  to both sides of the equation, we get

⇒ (x) 2  + 2 × x × 1/4 + (1/4) 2  = (1/4) 2  – 2

⇒ (x + 1/4) 2  = 1/16 – 2

⇒ (x + 1/4) 2  = -31/16

As we know, the square of numbers cannot be negative.

Therefore, there is no real root for the given equation, 2x 2 + x + 4 = 0

2. Find the roots of the quadratic equations given in Q.1 above by applying the quadratic formula.

(i) 2x 2  – 7x + 3 = 0

On comparing the given equation with ax 2  + bx + c = 0, we get,

a = 2, b = -7 and c = 3

By using the quadratic formula, we get,

⇒ x = (7±√(49 – 24))/4

⇒ x = (7±√25)/4

⇒ x = (7±5)/4

⇒ x = (7+5)/4 or x = (7-5)/4

⇒ x = 12/4 or 2/4

∴  x = 3 or 1/2

a = 2, b = 1 and c = -4

⇒x = (-1±√1+32)/4

⇒x = (-1±√33)/4

∴ x = (-1+√33)/4 or x = (-1-√33)/4

(iii) 4x 2  + 4√3x + 3 = 0

On comparing the given equation with ax 2  + bx + c = 0, we get

a = 4, b = 4√3 and c = 3

⇒ x = (-4√3±√48-48)/8

⇒ x = (-4√3±0)/8

∴ x = -√3/2 or x = -√3/2

(iv) 2x 2  + x + 4 = 0

a = 2, b = 1 and c = 4

By using the quadratic formula, we get

⇒ x = (-1±√1-32)/4

⇒ x = (-1±√-31)/4

As we know, the square of a number can never be negative. Therefore, there is no real solution for the given equation.

3. Find the roots of the following equations:

(i) x-1/x = 3, x ≠ 0 (ii) 1/x+4 – 1/x-7 = 11/30, x = -4, 7

(i) x-1/x = 3

⇒ x 2  – 3x -1 = 0

a = 1, b = -3 and c = -1

⇒ x = (3±√9+4)/2

⇒ x = (3±√13)/2

∴ x = (3+√13)/2 or x = (3-√13)/2

(ii) 1/x+4 – 1/x-7 = 11/30 ⇒ x-7-x-4/(x+4)(x-7) = 11/30

⇒ -11/(x+4)(x-7) = 11/30

⇒ (x+4)(x-7) = -30

⇒ x 2  – 3x – 28 = 30

⇒ x 2  – 3x + 2 = 0

We can solve this equation by factorisation method now.

⇒ x 2  – 2x – x + 2 = 0

⇒ x(x – 2) – 1(x – 2) = 0

⇒ (x – 2)(x – 1) = 0

⇒ x = 1 or 2

4. The sum of the reciprocals of Rehman’s age (in years) 3 years ago and 5 years from now is 1/3. Find his present age.

Let us say the present age of Rahman is x  years.

Three years ago, Rehman’s age was ( x  – 3) years.

Five years after, his age will be ( x  + 5) years.

Given the sum of the reciprocals of Rehman’s ages 3 years ago and after 5 years is equal to 1/3.

∴ 1/ x -3 + 1/ x -5 = 1/3

(x +5+ x -3)/( x -3)( x +5) = 1/3

(2 x +2)/( x -3)( x +5) = 1/3

⇒ 3(2 x  + 2) = ( x -3)( x +5)

⇒ 6 x  + 6 =  x 2  + 2 x  – 15

⇒  x 2  – 4 x  – 21 = 0

⇒  x 2  – 7 x  + 3 x  – 21 = 0

⇒  x ( x  – 7) + 3( x  – 7) = 0

⇒ ( x  – 7)( x  + 3) = 0

⇒  x  = 7, -3

As we know, age cannot be negative.

Therefore, Rahman’s present age is 7 years.

5. In a class test, the sum of Shefali’s marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210. Find her marks in the two subjects.

Let us say the marks of Shefali in Maths be x.

Then, the marks in English will be 30 – x.

As per the given question,

(x + 2)(30 – x – 3) = 210

(x + 2)(27 – x) = 210

⇒ -x 2  + 25x + 54 = 210

⇒ x 2  – 25x + 156 = 0

⇒ x 2  – 12x – 13x + 156 = 0

⇒ x(x – 12) -13(x – 12) = 0

⇒ (x – 12)(x – 13) = 0

⇒ x = 12, 13

Therefore, if the marks in Maths are 12, then marks in English will be 30 – 12 = 18, and if the marks in Maths are 13, then marks in English will be 30 – 13 = 17 .

6. The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field.

Let us say the shorter side of the rectangle is x  m.

Then, larger side of the rectangle = ( x  + 30) m

As given, the length of the diagonal is = x + 30 m

⇒  x 2  + ( x  + 30) 2  = ( x  + 60) 2

⇒  x 2  +  x 2  + 900 + 60 x  =  x 2  + 3600 + 120 x

⇒  x 2  – 60 x  – 2700 = 0

⇒  x 2  – 90 x  + 30 x  – 2700 = 0

⇒  x ( x  – 90) + 30( x  -90) = 0

⇒ ( x  – 90)( x  + 30) = 0

⇒  x  = 90, -30

However, the side of the field cannot be negative. Therefore, the length of the shorter side will be 90 m.

And the length of the larger side will be (90 + 30) m = 120 m.

7. The difference of the squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.

Let us say the larger and smaller number be x  and  y,  respectively.

As per the question given,

x 2  –  y 2  = 180 and  y 2  = 8 x

⇒  x 2  – 8 x  = 180

⇒  x 2  – 8 x  – 180 = 0

⇒  x 2  – 18 x  + 10 x  – 180 = 0

⇒  x ( x  – 18) +10( x  – 18) = 0

⇒ ( x  – 18)( x  + 10) = 0

⇒  x  = 18, -10

However, the larger number cannot be considered a negative number, as 8 times the larger number will be negative, and hence, the square of the smaller number will be negative, which is not possible.

Therefore, the larger number will be 18 only.

∴  y 2  = 8x = 8 × 18 = 144

⇒  y  = ±√144 = ±12

∴ Smaller number = ±12

Therefore, the numbers are 18 and 12 or 18 and -12.

8. A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.

It is given that

Distance = 360 km

Consider x as the speed, then the time taken

If the speed is increased by 5 km/h, the speed will be (x + 5) km/h.

Distance will be the same.

t = 360/(x + 5)

We know that

Time with original speed – Time with increased speed = 1

360/x – 360/(x + 5) = 1

LCM = x (x + 5)

360 x + 1800 – 360x = x (x + 5)

x 2  + 5x = 1800

x 2  + 5x – 1800 = 0

x 2  + 45x – 40x – 1800 = 0

x (x + 45) – 40 (x + 45) = 0

(x – 40) (x + 45) = 0

x = 40 km/hr

As we know, the value of speed cannot be negative.

Therefore, the speed of the train is 40 km/h.

Let the time taken by the smaller pipe to fill the tank = x hr.

Time taken by the larger pipe = ( x  – 10) hr

Part of the tank filled by smaller pipe in 1 hour = 1/ x

Part of the tank filled by larger pipe in 1 hour = 1/( x  – 10)

1/ x  + 1/ x -10 = 8/75

x -10+ x / x ( x -10) = 8/75

⇒ 2 x -10/ x ( x -10) = 8/75

⇒ 75(2 x  – 10) = 8 x 2  – 80 x

⇒ 150 x  – 750 = 8 x 2  – 80 x

⇒ 8 x 2  – 230 x  +750 = 0

⇒ 8 x 2  – 200 x  – 30 x  + 750 = 0

⇒ 8 x ( x  – 25) -30( x  – 25) = 0

⇒ ( x  – 25)(8 x  -30) = 0

⇒  x  = 25, 30/8

Time taken by the smaller pipe cannot be 30/8 = 3.75 hours, as the time taken by the larger pipe will become negative, which is logically not possible.

Therefore, the time taken individually by the smaller pipe and the larger pipe will be 25 and 25 – 10 =15 hours, respectively.

10. An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is 11 km/h more than that of the passenger train, find the average speed of the two trains.

Let us say the average speed of the passenger train =   x  km/h.

Average speed of express train = ( x  + 11) km/h

Given the time taken by the express train to cover 132 km is 1 hour less than the passenger train to cover the same distance. Therefore,

(132/x) – (132/(x+11)) = 1

132(x+11-x)/(x(x+11)) = 1

132 × 11 /(x(x+11)) = 1

⇒ 132 × 11 =  x ( x  + 11)

⇒  x 2  + 11 x  – 1452 = 0

⇒  x 2  +  44 x  -33 x  -1452 = 0

⇒  x ( x  + 44) -33( x  + 44) = 0

⇒ ( x  + 44)( x  – 33) = 0

⇒  x  = – 44, 33

As we know, speed cannot be negative.

Therefore, the speed of the passenger train will be 33 km/h and thus, the speed of the express train will be 33 + 11 = 44 km/h.

11. Sum of the areas of two squares is 468 m 2 . If the difference between their perimeters is 24 m, find the sides of the two squares.

Let the sides of the two squares be  x  m and  y  m.

Therefore, their perimeter will be 4 x  and 4 y,  respectively

And the area of the squares will be x 2  and  y 2,  respectively.

4 x  – 4 y  = 24

x  –  y  = 6

x  =  y  + 6

Also,  x 2  +   y 2  = 468

⇒ (6 +  y 2 ) +  y 2  = 468

⇒ 36 +  y 2  + 12 y  +  y 2  = 468

⇒ 2 y 2  + 12 y  + 432 = 0

⇒  y 2  + 6y – 216 = 0

⇒  y 2  + 18 y  – 12 y  – 216 = 0

⇒  y ( y  +18) -12( y  + 18) = 0

⇒ ( y  + 18)( y  – 12) = 0

⇒  y  = -18, 12

As we know, the side of a square cannot be negative.

Hence, the sides of the squares are 12 m and (12 + 6) m = 18 m.

Exercise 4.4 Page: 91

1. Find the nature of the roots of the following quadratic equations. If the real roots exist, find them. (i) 2 x 2  – 3 x  + 5 = 0 (ii) 3 x 2  – 4√3 x  + 4 = 0 (iii) 2 x 2  – 6 x  + 3 = 0

2x 2  – 3 x  + 5 = 0

Comparing the equation with  ax 2  +  bx  +  c  = 0, we get

a  = 2,  b  = -3 and  c  = 5

We know, Discriminant =  b 2  – 4 ac

=  ( – 3) 2  – 4 (2) (5) = 9 – 40

As you can see, b 2  – 4ac < 0

Therefore, no real root is possible for the given equation, 2x 2  – 3 x + 5 = 0

(ii) 3 x 2  – 4√3 x  + 4 = 0

a  = 3,  b  = -4√3 and  c  = 4

= (-4√3) 2  – 4(3)(4)

= 48 – 48 = 0

As  b 2  – 4 ac  = 0,

Real roots exist for the given equation, and they are equal to each other.

Hence, the roots will be – b /2 a  and – b /2 a .

– b /2 a  = -(-4√3)/2×3 = 4√3/6 = 2√3/3 = 2/√3

Therefore, the roots are 2/√3 and 2/√3.

(iii) 2 x 2  – 6 x  + 3 = 0

a  = 2,  b  = -6,  c  = 3

As we know, Discriminant =  b 2  – 4 ac

= (-6) 2  – 4 (2) (3)

= 36 – 24 = 12

As  b 2  – 4 ac  > 0,

Therefore, there are distinct real roots that exist for this equation, 2 x 2  – 6 x + 3 = 0

= (-(-6) ± √(-6 2 -4(2)(3)) )/ 2(2)

= (6±2√3 )/4

Therefore, the roots for the given equation are (3+√3)/2 and (3-√3)/2

2. Find the values of  k for each of the following quadratic equations so that they have two equal roots. (i) 2 x 2  +  kx  + 3 = 0 (ii)  kx  ( x  – 2) + 6 = 0

(i) 2 x 2  +  kx  + 3 = 0

Comparing the given equation with  ax 2  +  bx  +  c  = 0, we get,

a  = 2,  b  = k and  c  = 3

= ( k ) 2  – 4(2) (3)

=  k 2  – 24

For equal roots, we know,

Discriminant = 0

k 2  – 24 = 0

k = ±√24 = ±2√6

(ii)  kx ( x  – 2) + 6 = 0

or  kx 2  – 2 kx  + 6 = 0

Comparing the given equation with  ax 2  +  bx  +  c  = 0, we get

a  =  k ,  b  = – 2 k  and  c  = 6

= ( – 2 k ) 2 – 4 ( k ) (6)

= 4 k 2 – 24 k

b 2 – 4 ac  = 0

4 k 2 – 24 k  = 0

4 k  ( k  – 6) = 0

Either 4 k  = 0 or  k  = 6 = 0

k  = 0 or  k  = 6

However, if  k  = 0, then the equation will not have the terms ‘ x 2 ‘ and ‘ x ‘.

Therefore, if this equation has two equal roots,  k  should be 6 only.

3. Is it possible to design a rectangular mango grove whose length is twice its breadth and the area is 800 m 2 ? If so, find its length and breadth.

Let the breadth of the mango grove be l .

The length of the mango grove will be 2 l .

Area of the mango grove = (2 l ) ( l )= 2 l 2

2 l 2  = 800

l 2  = 800/2 = 400

l 2  – 400 =0

a  = 1,  b  = 0,  c  = 400

=> (0) 2  – 4 × (1) × ( – 400) = 1600

Here,  b 2  – 4 ac  > 0

Thus, the equation will have real roots. And hence, the desired rectangular mango grove can be designed.

As we know, the value of length cannot be negative.

Therefore, the breadth of the mango grove = 20 m

Length of mango grove = 2 × 20 = 40 m

4. Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is 20 years. Four years ago, the product of their age in years was 48.

Let’s say the age of one friend is x years.

Then, the age of the other friend will be (20 – x) years.

Four years ago,

Age of First friend = ( x  – 4) years

Age of Second friend = (20 –  x  – 4) = (16 –  x ) years

As per the given question, we can write,

( x  – 4) (16 –  x ) = 48

16 x – x 2  – 64 + 4 x  = 48

 – x 2  + 20 x –  112 = 0

x 2  – 20 x +  112 = 0

a  =  1 ,  b  = -2 0  and  c  = 112

Discriminant =  b 2  – 4 ac

= (- 20 ) 2  – 4 × 112

= 400 – 448 = -48

b 2  – 4 ac  < 0

Therefore, there will be no real solution possible for the equations. Hence, the condition doesn’t exist.

5. Is it possible to design a rectangular park of perimeter 80 and an area of 400 m2? If so, find its length and breadth.

Let the length and breadth of the park be l and b.

Perimeter of the rectangular park = 2 ( l + b ) = 80

So, l + b  = 40

Or,  b  = 40 –  l

Area of the rectangular park =  l×b = l(40 – l) =  40 l  –  l 2 = 400

l 2   –    40 l  + 400   = 0, which is a quadratic equation.

a  = 1,  b  = -40,  c  = 400

Since, Discriminant =  b 2  – 4 ac

=(- 40 ) 2  – 4 × 400

= 1600 – 1600 = 0

Thus, b 2  – 4 ac  = 0

Therefore, this equation has equal real roots. Hence, the situation is possible.

The root of the equation,

l  = – b /2 a

l  = -(-40)/2(1) = 40/2 = 20

Therefore, the length of the rectangular park, l  = 20 m

And the breadth of the park, b  = 40 –  l  = 40 – 20 = 20 m.

NCERT Solutions for Class 10 Maths Chapter 4 – Quadratic Equations

A 1-mark question was asked from Chapter 4 Quadratic Equations in the year 2018. However, in the year 2017, a total of 13 marks were asked from the topic Quadratic Equations. Therefore, students need to have a thorough understanding of the topic. The topics and sub-topics provided in this chapter include:

4.1 Introduction

If we equate the polynomial  ax 2 + bx + c, a ≠ 0 to zero, we get a quadratic equation. Quadratic equations come up when we deal with many real-life situations. In this chapter, students will study quadratic equations and various ways of finding their roots. They will also see some applications of quadratic equations in daily life situations.

4.2 Quadratic Equations

A quadratic equation in the variable x is an equation of the form ax 2 + bx + c = 0, where a, b, c are real numbers, a ≠ 0. In fact, any equation of the form p(x) = 0, where p(x) is a polynomial of degree 2, is a quadratic equation. But when we write the terms of p(x) in descending order of their degrees, then we get the standard form of the equation. That is, ax 2 + bx + c = 0, a ≠ 0 is called the standard form of a quadratic equation.

4.3 Solution of Quadratic Equations by Factorisation

A real number α is called a root of the quadratic equation ax 2  + bx + c = 0, a ≠ 0 if a α 2 + bα + c = 0. We also say that x = α is a solution of the quadratic equation or that α satisfies the quadratic equation. Note that the zeroes of the quadratic polynomial ax 2 + bx + c and the roots of the quadratic equation ax 2  + bx + c = 0 are the same.

4.4 Solution of a Quadratic Equation by Completing the Square

Finding the value that makes a quadratic equation a square trinomial is called  completing the square. The square trinomial can then be solved easily by factorising.

4.5 Nature of Roots

If b 2  – 4ac < 0, then there is no real number whose square is b 2  – 4ac. Therefore, there are no real roots for the given quadratic equation in this case. Since b 2  – 4ac determines whether the quadratic equation ax 2  + bx + c = 0 has real roots or not, b 2  – 4ac is called the discriminant of this quadratic equation. So, a quadratic equation ax 2  + bx + c = 0 has (i) two distinct real roots, if b 2  – 4ac > 0, (ii) two equal real roots, if b 2 – 4ac = 0, (iii) no real roots, if b 2 – 4ac < 0. 4.6

List of exercises we covered in NCERT Solutions for Class 10 Maths Chapter 4

Exercise 4.1 Solutions – 2 Questions Exercise 4.2 Solutions – 6 Questions Exercise 4.3 Solutions – 11 Questions Exercise 4.4 Solutions – 5 Questions

In a quadratic equation, x represents an unknown form, and a, b, and c are the known values. An equation to be quadratic “a” should not be equal to 0. The equation is of the form ax 2 + bx + c = 0. The values of a, b, and c are always real numbers. A quadratic equation can be calculated by completing the square. A quadratic equation has

• Two different real roots
• No real roots
• Two equal roots

Key Features of NCERT Solutions for Class 10 Maths Chapter 4 – Quadratic Equations

• The highly experienced faculty at BYJU’S designs NCERT Solutions with utmost care.
• The solutions are 100% accurate and can be used by the students while preparing for their CBSE board exams.
• All the minute concepts are also covered to help students face other competitive exams more confidently.
• The exercise questions present in the NCERT Textbook have been answered in a step-wise manner so that students attain good scores not only on the final answer but also on each step.

For more questions to practise, students can refer to the other study materials which are given at BYJU’S.

• RD Sharma Solutions for Class 10 Maths Chapter 8 Quadratic Equations

Disclaimer – 

Dropped Topics –  4.4 Solution of a quadratic equation by completing the squares

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4.4E: Exercises - Graphing Quadratic Functions Using Properties

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Practice Makes Perfect

Exercises 1 - 4: recognize the graph of a quadratic function.

In the following exercises, graph the functions by plotting points.

1. \(f(x)=x^{2}+3\)

2. \(f(x)=x^{2}-3\)

3. \(y=-x^{2}+1\)

4. \(f(x)=-x^{2}-1\)

Exercises 5 - 8: Recognize the Graph of a Quadratic Function

For each of the following exercises, determine if the parabola opens up or down.

5. a. \(f(x)=-2 x^{2}-6 x-7\) b. \(f(x)=6 x^{2}+2 x+3\)

6. a. \(f(x)=4 x^{2}+x-4\) b. \(f(x)=-9 x^{2}-24 x-16\)

7. a. \(f(x)=-3 x^{2}+5 x-1\) b. \(f(x)=2 x^{2}-4 x+5\)

8. a. \(f(x)=x^{2}+3 x-4\) b. \(f(x)=-4 x^{2}-12 x-9\)

5. a. down b. up

7. a. down b. up

Exercises 9 - 12: Find the Axis of Symmetry and Vertex of a Parabola

In the following functions, find

• The equation of the axis of symmetry
• The vertex of its graph

9. \(f(x)=x^{2}+8 x-1\)

10. \(f(x)=x^{2}+10 x+25\)

11. \(f(x)=-x^{2}+2 x+5\)

12. \(f(x)=-2 x^{2}-8 x-3\)

9. a. Axis of symmetry: \(x=-4\) b. Vertex: \((-4,-17)\)

11. a. Axis of symmetry: \(x=1\) b. Vertex: \((1,2)\)

Exercises 13 - 24: Find the Intercepts of a Parabola

In the following exercises, find the intercepts of the parabola whose function is given.

13. \(f(x)=x^{2}+7 x+6\)

14. \(f(x)=x^{2}+10 x-11\)

15. \(f(x)=x^{2}+8 x+12\)

16. \(f(x)=x^{2}+5 x+6\)

17. \(f(x)=-x^{2}+8 x-19\)

18. \(f(x)=-3 x^{2}+x-1\)

19. \(f(x)=x^{2}+6 x+13\)

20. \(f(x)=x^{2}+8 x+12\)

21. \(f(x)=4 x^{2}-20 x+25\)

22. \(f(x)=-x^{2}-14 x-49\)

23. \(f(x)=-x^{2}-6 x-9\)

24. \(f(x)=4 x^{2}+4 x+1\)

13. \(y\)-intercept: \((0,6)\); \(x\)-intercept(s): \((-1,0), (-6,0)\)

15. \(y\)-intercept: \((0,12)\); \(x\)-intercept(s): \((-2,0), (-6,0)\)

17. \(y\)-intercept: \((0,-19)\); \(x\)-intercept(s): none

19. \(y\)-intercept: \((0,13)\); \(x\)-intercept(s): none

21. \(y\)-intercept: \((0,-16)\); \(x\)-intercept(s): \((\frac{5}{2},0)\)

23. \(y\)-intercept: \((0,9)\); \(x\)-intercept(s): \((-3,0)\)

Exercises 25 - 42: Graph Quadratic Functions Using Properties

In the following exercises, graph the function by using its properties.

25. \(f(x)=x^{2}+6 x+5\)

26. \(f(x)=x^{2}+4 x-12\)

27. \(f(x)=x^{2}+4 x+3\)

28. \(f(x)=x^{2}-6 x+8\)

29. \(f(x)=9 x^{2}+12 x+4\)

30. \(f(x)=-x^{2}+8 x-16\)

31. \(f(x)=-x^{2}+2 x-7\)

32. \(f(x)=5 x^{2}+2\)

33. \(f(x)=2 x^{2}-4 x+1\)

34. \(f(x)=3 x^{2}-6 x-1\)

35. \(f(x)=2 x^{2}-4 x+2\)

36. \(f(x)=-4 x^{2}-6 x-2\)

37. \(f(x)=-x^{2}-4 x+2\)

38. \(f(x)=x^{2}+6 x+8\)

39. \(f(x)=5 x^{2}-10 x+8\)

40. \(f(x)=-16 x^{2}+24 x-9\)

41. \(f(x)=3 x^{2}+18 x+20\)

42. \(f(x)=-2 x^{2}+8 x-10\)

Exercises 43 - 48: Solve Maximum and Minimum Applications

In the following exercises, find the maximum or minimum value of each function.

43. \(f(x)=2 x^{2}+x-1\)

44. \(y=-4 x^{2}+12 x-5\)

45. \(y=x^{2}-6 x+15\)

46. \(y=-x^{2}+4 x-5\)

47. \(y=-9 x^{2}+16\)

48. \(y=4 x^{2}-49\)

43. The minimum value is \(−\frac{9}{8}\) when \(x=−\frac{1}{4}\).

45. The maximum value is \(6\) when \(x=3\).

47. The maximum value is \(16\) when \(x=0\).

Exercises 49 - 60: Solve Maximum and Minimum Applications

In the following exercises, solve. Round answers to the nearest tenth.

49. An arrow is shot vertically upward from a platform \(45\) feet high at a rate of \(168\) ft/sec. Use the quadratic function \(h(t)=-16 t^{2}+168 t+45\) find how long it will take the arrow to reach its maximum height, and then find the maximum height.

50. A stone is thrown vertically upward from a platform that is \(20\) feet height at a rate of \(160\) ft/sec. Use the quadratic function \(h(t)=-16 t^{2}+160 t+20\) to find how long it will take the stone to reach its maximum height, and then find the maximum height.

51. A ball is thrown vertically upward from the ground with an initial velocity of \(109\) ft/sec. Use the quadratic function \(h(t)=-16 t^{2}+109 t+0\) to find how long it will take for the ball to reach its maximum height, and then find the maximum height.

52. A ball is thrown vertically upward from the ground with an initial velocity of \(122\) ft/sec. Use the quadratic function \(h(t)=-16 t^{2}+122 t+0\) to find how long it will take for the ball to reach its maximum height, and then find the maximum height.

53. A computer store owner estimates that by charging \(x\) dollars each for a certain computer, he can sell \(40 − x\) computers each week. The quadratic function \(R(x)=-x^{2}+40 x\) is used to find the revenue, \(R\), received when the selling price of a computer is \(x\), Find the selling price that will give him the maximum revenue, and then find the amount of the maximum revenue.

54. A retailer who sells backpacks estimates that by selling them for \(x\) dollars each, he will be able to sell \(100 − x\) backpacks a month. The quadratic function \(R(x)=-x^{2}+100 x\) is used to find the \(R\), received when the selling price of a backpack is \(x\). Find the selling price that will give him the maximum revenue, and then find the amount of the maximum revenue.

55. A retailer who sells fashion boots estimates that by selling them for \(x\) dollars each, he will be able to sell \(70 − x\) boots a week. Use the quadratic function \(R(x)=-x^{2}+70 x\) to find the revenue received when the average selling price of a pair of fashion boots is \(x\). Find the selling price that will give him the maximum revenue, and then find the amount of the maximum revenue per day.

56. A cell phone company estimates that by charging \(x\) dollars each for a certain cell phone, they can sell \(8 − x\) cell phones per day. Use the quadratic function \(R(x)=-x^{2}+8 x\) to find the revenue received per day when the selling price of a cell phone is \(x\). Find the selling price that will give them the maximum revenue per day, and then find the amount of the maximum revenue.

57. A rancher is going to fence three sides of a corral next to a river. He needs to maximize the corral area using \(240\) feet of fencing. The quadratic equation \(A(x)=x(240-2 x)\) gives the area of the corral, \(A\), for the length, \(x\), of the corral along the river. Find the length of the corral along the river that will give the maximum area, and then find the maximum area of the corral.

58. A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using \(100\) feet of fencing. The quadratic function \(A(x)=x(100-2 x)\) gives the area, \(A\), of the dog run for the length, \(x\), of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.

59. A land owner is planning to build a fenced in rectangular patio behind his garage, using his garage as one of the “walls.” He wants to maximize the area using \(80\) feet of fencing. The quadratic function \(A(x)=x(80-2 x)\) gives the area of the patio, where \(x\) is the width of one side. Find the maximum area of the patio.

60. A family of three young children just moved into a house with a yard that is not fenced in. The previous owner gave them \(300\) feet of fencing to use to enclose part of their backyard. Use the quadratic function \(A(x)=x(300-2 x)\) to determine the maximum area of the fenced in yard.

49. In \(5.3\) sec the arrow will reach maximum height of \(486\) ft.

51. In \(3.4\) seconds the ball will reach its maximum height of \(185.6\) feet.

53. \(20\) computers will give the maximum of $\(400\) in receipts.

55. He will be able to sell \(35\) pairs of boots at the maximum revenue of $\(1,225\).

57. The length of the side along the river of the corral is \(120\) feet and the maximum area is \(7,200\) square feet.

59. The maximum area of the patio is \(800\) feet.

Exercises 61 - 64: Writing Exercises

61. How do the graphs of the functions \(f(x)=x^{2}\) and \(f(x)=x^{2}−1\) differ? We graphed them at the start of this section. What is the difference between their graphs? How are their graphs the same?

62. Explain the process of finding the vertex of a parabola.

63. Explain how to find the intercepts of a parabola.

64. How can you use the discriminant when you are graphing a quadratic function?

1. Answers will vary.

3. Answers will vary.

a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

b. After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?