Solving Quadratic Equations by the Quadratic Formula

The completing square method is still a long method for solving the quadratic equation, so to make calculations shorter and easier, a formula was developed by mathematicians to solve the quadratic equation. It is called the quadratic formula.

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

Write the equation in standard form
a{x^2} + bx + c = 0 where a \ne 0

Shifting the constant term on RHS, we get
a{x^2} + bx = - c

Make the coefficient of {x^2} as 1 and divide the equation by a
{x^2} + \frac{b}{a}x = - \frac{c}{a}

Adding {\left( {\frac{b}{{2a}}} \right)^2} to 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 the square root of 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 the 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 the 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