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Key Questions
A few examples...
Explanation:
I will assume that you mean things like common identities and the quadratic formula . Here are just a few:
Difference of squares identity
#a^2-b^2 = (a-b)(a+b)#
Deceptively simple, but massively useful.
For example:
#a^4+b^4 = (a^2+b^2)^2 - 2a^2b^2# #color(white)(a^4+b^4) = (a^2+b^2)^2 - (sqrt(2)ab)^2# #color(white)(a^4+b^4) = ((a^2+b^2) - sqrt(2)ab)((a^2+b^2) +sqrt(2)ab)# #color(white)(a^4+b^4) = (a^2-sqrt(2)ab+b^2)(a^2+sqrt(2)ab+b^2)#
Difference of cubes identity
#a^3-b^3 = (a-b)(a^2+ab+b^2)#
Sum of cubes identity
#a^3+b^3 = (a+b)(a^2-ab+b^2)#
Quadratic formula
Very useful to know, better if you know how to derive it:
The zeros of #ax^2+bx+c# are given by:
#x = (-b+-sqrt(b^2-4ac))/(2a)#
Pythagoras theorem
If a right angled triangle has legs of length #a, b# and hypotenuse of length #c# then:
#c^2 = a^2+b^2#
This is also very useful in trigonometric form. If we have an angle #theta# in a right-angled triangle, then we call the side nearest #theta# , the #"adjacent"# side, the side opposite it the #"opposite"# side and the hypotenuse the #"hypotenuse"# .
#"hypotenuse"^2 = "adjacent"^2 + "opposite"^2#
Dividing both sides by #"hypotenuse"^2# , we get:
#1 = ("adjacent"/"hypotenuse")^2 + ("opposite"/"hypotenuse")^2#
#1 = cos^2 theta + sin^2 theta#
Then dividing both sides by #cos^2 theta# we find:
#sec^2 theta = 1 + tan^2 theta#
Binomial theorem
#(a+b)^n = sum_(k=0)^n ((n), (k)) a^(n-k) b^k#
where #((n), (k)) = (n!)/((n-k)! k!)#
#(x+1)^4 = x^4+4x^3+6x^2+4x+1#
Using Pythagoras' Theorem, we know #A^2 + B^2 = C^2#
#A^2 = 9# , #B^2 = 16 implies A^2 + B^2 = 9 + 16 = 25#
SInce #C^2 = A^2 + B^2 = 25# , you know that
#C = sqrt(C^2) = sqrt(25) = 5#
You simply find out what value goes with what part of the formula, and do this for each, and then work out the formula as normal.
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