# Is there an easy way of remembering the quadratic formula?

May 3, 2018

Quick writing

#### Explanation:

Here's someone else's mnemonic:
From square of b, take 4ac;
Square root extract, and b subtract;
Divide by 2a; you’ve x, hooray!

May 3, 2018

Here's what I think is the best way...

#### Explanation:

The very best way to remember it is to learn how to derive it.

We will complete the square and use the difference of squares identity:

${A}^{2} - {B}^{2} = \left(A - B\right) \left(A + B\right)$

with $A = 2 a x + b$ and $B = \sqrt{{b}^{2} - 4 a c}$

Given:

$a {x}^{2} + b x + c = 0 \text{ }$ with $\setminus a \ne 0$

Multiply by $4 a$ to get:

$0 = 4 {a}^{2} {x}^{2} + 4 a b x + 4 a c$

$\textcolor{w h i t e}{0} = {\left(2 a x\right)}^{2} + 2 b \left(2 a x\right) + 4 a c$

$\textcolor{w h i t e}{0} = {\left(2 a x\right)}^{2} + 2 b \left(2 a x\right) + {b}^{2} + 4 a c - {b}^{2}$

$\textcolor{w h i t e}{0} = {\left(2 a x + b\right)}^{2} - \left({b}^{2} - 4 a c\right)$

$\textcolor{w h i t e}{0} = {\left(2 a x + b\right)}^{2} - {\left(\sqrt{{b}^{2} - 4 a c}\right)}^{2}$

$\textcolor{w h i t e}{0} = \left(\left(2 a x + b\right) - \sqrt{{b}^{2} - 4 a c}\right) \left(\left(2 a x + b\right) + \sqrt{{b}^{2} - 4 a c}\right)$

$\textcolor{w h i t e}{0} = \left(2 a x + b - \sqrt{{b}^{2} - 4 a c}\right) \left(2 a x + b + \sqrt{{b}^{2} - 4 a c}\right)$

Hence:

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

Then dividing both sides by $2 a$, we get:

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