What are commonly used formulas used in problem solving?

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

${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$

Deceptively simple, but massively useful.

For example:

${a}^{4} + {b}^{4} = {\left({a}^{2} + {b}^{2}\right)}^{2} - 2 {a}^{2} {b}^{2}$

$\textcolor{w h i t e}{{a}^{4} + {b}^{4}} = {\left({a}^{2} + {b}^{2}\right)}^{2} - {\left(\sqrt{2} a b\right)}^{2}$

$\textcolor{w h i t e}{{a}^{4} + {b}^{4}} = \left(\left({a}^{2} + {b}^{2}\right) - \sqrt{2} a b\right) \left(\left({a}^{2} + {b}^{2}\right) + \sqrt{2} a b\right)$

$\textcolor{w h i t e}{{a}^{4} + {b}^{4}} = \left({a}^{2} - \sqrt{2} a b + {b}^{2}\right) \left({a}^{2} + \sqrt{2} a b + {b}^{2}\right)$

Difference of cubes identity

${a}^{3} - {b}^{3} = \left(a - b\right) \left({a}^{2} + a b + {b}^{2}\right)$

Sum of cubes identity

${a}^{3} + {b}^{3} = \left(a + b\right) \left({a}^{2} - a b + {b}^{2}\right)$

Very useful to know, better if you know how to derive it:

The zeros of $a {x}^{2} + b x + c$ are given by:

$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

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 $\text{adjacent}$ side, the side opposite it the $\text{opposite}$ side and the hypotenuse the $\text{hypotenuse}$.

Then:

${\text{hypotenuse"^2 = "adjacent"^2 + "opposite}}^{2}$

Dividing both sides by ${\text{hypotenuse}}^{2}$, we get:

$1 = {\left(\text{adjacent"/"hypotenuse")^2 + ("opposite"/"hypotenuse}\right)}^{2}$

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

${\left(a + b\right)}^{n} = {\sum}_{k = 0}^{n} \left(\begin{matrix}n \\ k\end{matrix}\right) {a}^{n - k} {b}^{k}$

where ((n), (k)) = (n!)/((n-k)! k!)

For example:

${\left(x + 1\right)}^{4} = {x}^{4} + 4 {x}^{3} + 6 {x}^{2} + 4 x + 1$