# Solving Quadratic Equations by Quadratic Formula

The method of completing square is still a long method for solving the quadratic equation, so to make calculation further short and easier, a formula is developed by the mathematician to solve quadratic equation, called quadratic formula.

In order to derive quadratic formula, the method of completing square is used. This is given below

Write the equation in standard form
$a{x^2} + bx + c = 0$        where $a \ne 0$
Shift constant term on RHS, we get
$a{x^2} + bx = - c$
Make the coefficient of ${x^2}$ as $1$ divide the equation by $a$
${x^2} + \frac{b}{a}x = - \frac{c}{a}$
Adding ${\left( {\frac{b}{{2a}}} \right)^2}$ on both sides, we get
${x^2} + \frac{b}{a}x + {\left( {\frac{b}{{2a}}} \right)^2} = {\left( {\frac{b}{{2a}}} \right)^2} - \frac{c}{a}$
${\left( {x + \frac{b}{{2a}}} \right)^2} = \frac{{{b^2}}}{{4{a^2}}} - \frac{c}{a}$
${\left( {x + \frac{b}{{2a}}} \right)^2} = \frac{{{b^2} - 4ac}}{{4{a^2}}}$
Taking square root both sides, we get
$\sqrt {{{\left( {x + \frac{b}{{2a}}} \right)}^2}} = \pm \sqrt {\frac{{{b^2} - 4ac}}{{4{a^2}}}}$
$x + \frac{b}{{2a}} = \pm \sqrt {\frac{{{b^2} - 4ac}}{{4{a^2}}}}$
$x = - \frac{b}{{2a}} \pm \frac{{\sqrt {{b^2} - 4ac} }}{{2a}}$
$\boxed{x = \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}}$
This is the quadratic formula.

Example:

Solve the equation by quadratic formula
$16 + {x^2} - 10x = 0$
Solution:
Write the equation in standard form
${x^2} - 10x + 16 = 0$
Let$a = 1$, $b = - 10$, $c = 16$
Using quadratic formula, we get
$x = \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
$x = \frac{{ - \left( { - 10} \right) \pm \sqrt {{{\left( { - 10} \right)}^2} - 4\left( 1 \right)\left( {16} \right)} }}{{2\left( 1 \right)}}$
$x = \frac{{10 \pm \sqrt {100 - 64} }}{2} = \frac{{10 \pm \sqrt {36} }}{2}$
$x = \frac{{10 \pm 6}}{2}$
Either              $x = \frac{{10 + 6}}{2}$   or   $x = \frac{{10 - 6}}{2}$
$x = \frac{{16}}{2} = 8$          $x = \frac{4}{2} = 2$