Question

What are commonly used formulas used in problem solving?

Answer 1

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`