# What is the improved quadratic formula to solve quadratic equations ?

Jan 1, 2018

There is only one quadratic formula, that is $x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$.

#### Explanation:

For a general solution of $x$ in $a {x}^{2} + b x + c = 0$, we can derive the quadratic formula $x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$.

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

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

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

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

Now, you can factorize.
${\left(2 a x + b\right)}^{2} = {b}^{2} - 4 a c$

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

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

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

Jan 3, 2018

This could refer to...

#### Explanation:

One of the nuisances when using the quadratic formula is that often the square root can be simplified, involving at least one more step than necessary. If the middle coefficient is even, then we can avoid this by using an alternative formulation of the quadratic formula.

Given:

$a {x}^{2} + 2 \mathrm{dx} + c = 0$

The roots are given by the formula:

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