# What is the quadratic formula?

Mar 26, 2018

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

#### Explanation:

Negative b plus minus the square root of b squared minus 4*a*c over 2*a. To plug something into the quadratic formula the equation needs to be in standard form ($a {x}^{2} + b {x}^{2} + c$).

hope this helps!

Mar 27, 2018

If we have:

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

Then:

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

#### Explanation:

The quadratic formula provides a method of solving a generic quadratic equation:

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

To solve the equation we first factor out $a$:

$a \left\{{x}^{2} + \frac{b}{a} x + \frac{c}{a}\right\} = 0 \implies {x}^{2} + \frac{b}{a} x + \frac{c}{a} = 0$

Then we complete the square:

${\left(x + \frac{b}{2 a}\right)}^{2} - {\left(\frac{b}{2 a}\right)}^{2} + \frac{c}{a} = 0$

Now, we solve for $x$:

${\left(x + \frac{b}{2 a}\right)}^{2} = {\left(\frac{b}{2 a}\right)}^{2} - \frac{c}{a}$
$\text{ } = {b}^{2} / \left(4 {a}^{2}\right) - \frac{c}{a}$

$\text{ } = {b}^{2} / \left(4 {a}^{2}\right) - \frac{4 a c}{4 {a}^{2}}$

$\text{ } = \frac{{b}^{2} - 4 a c}{4 {a}^{2}}$

By taking square root we get:

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

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

So that:

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

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

Which is known as the "quadratic formula".