What are commonly used formulas used in problem solving?
1 Answer
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
#x = (-b+-sqrt(b^2-4ac))/(2a)#
Pythagoras theorem
If a right angled triangle has legs of length
#c^2 = a^2+b^2#
This is also very useful in trigonometric form. If we have an angle
Then:
#"hypotenuse"^2 = "adjacent"^2 + "opposite"^2#
Dividing both sides by
#1 = ("adjacent"/"hypotenuse")^2 + ("opposite"/"hypotenuse")^2#
That is:
#1 = cos^2 theta + sin^2 theta#
Then dividing both sides by
#sec^2 theta = 1 + tan^2 theta#
Binomial theorem
#(a+b)^n = sum_(k=0)^n ((n), (k)) a^(n-k) b^k#
where
For example:
#(x+1)^4 = x^4+4x^3+6x^2+4x+1#