What are commonly used formulas used in problem solving?

1 Answer
Feb 10, 2018

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".

Then:

"hypotenuse"^2 = "adjacent"^2 + "opposite"^2

Dividing both sides by "hypotenuse"^2, we get:

1 = ("adjacent"/"hypotenuse")^2 + ("opposite"/"hypotenuse")^2

That is:

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!)

For example:

(x+1)^4 = x^4+4x^3+6x^2+4x+1