Completing the Square

Key Questions

• Remember that

${\left(a x + b\right)}^{2} = {a}^{2} {x}^{2} + 2 a b x + {b}^{2}$

So the general idea to get ${b}^{2}$ is by getting $x$'s coefficient,
dividing by $2 a$, and squaring the result.

Example

$3 {x}^{2} + 12 x$

$a = 3$
$2 a b = 12$
$6 b = 12$
$b = 2$
$3 {x}^{2} + 12 x + 4 = {\left(3 x + 2\right)}^{2}$

You can also factor out ${x}^{2}$'s coefficient.. and proceed with completing the square.

Example:
$2 {X}^{2} + 4 X$

$2 \left({X}^{2} + 2 X\right)$

$2 {\left(X + 1\right)}^{2}$

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

The secret is that $\frac{b}{2 a}$ bit

Explanation:

Suppose you are given a quadratic equation to solve:

$2 {x}^{2} - 3 x - 2 = 0$

..which is in the form..

$a {x}^{2} + b x + c = 0$ with $a = 2$, $b = - 3$ and $c = - 2$

$\frac{b}{2 a} = - \frac{3}{4}$

So we find:

$2 {\left(x - \frac{3}{4}\right)}^{2} = 2 \left({x}^{2} - \left(2 \cdot x \cdot \frac{3}{4}\right) + {\left(\frac{3}{4}\right)}^{2}\right)$

$= 2 \left({x}^{2} - \frac{3 x}{2} + \frac{9}{16}\right)$

$= 2 {x}^{2} - 3 x + \frac{9}{8}$

So:

$2 {\left(x - \frac{3}{4}\right)}^{2} - \frac{25}{8} = 2 {\left(x - \frac{3}{4}\right)}^{2} - \frac{9}{8} - 2$

$= 2 {x}^{2} - 3 x + \frac{9}{8} - \frac{9}{8} - 2$

$= 2 {x}^{2} - 3 x - 2$

So:

$2 {x}^{2} - 3 x - 2 = 0$

turns into:

$2 {\left(x - \frac{3}{4}\right)}^{2} - \frac{25}{8} = 0$

Hence:

${\left(x - \frac{3}{4}\right)}^{2} = \frac{25}{16}$

So:

$x - \frac{3}{4} = \pm \sqrt{\frac{25}{16}} = \pm \frac{5}{4}$

and

$x = \frac{3}{4} \pm \frac{5}{4}$

In order to use the Completing the Square method, the value for $a$ in the quadratic equation must be $1$. If it is not $1$, you will have to use the AC method or the quadratic formula in order to solve for $x$.

Explanation:

Completing the square is a method used to solve a quadratic equation, $a {x}^{2} + b x + c$, where $a$ must be $1$. The goal is to force a perfect square trinomial on one side and then solving for $x$ by taking the square root of both sides.

The method is explained at the following website:

http://www.regentsprep.org/regents/math/algtrig/ate12/completesqlesson.htm

• Completing the square is a method that represents a quadratic equation as a combination of quadrilateral used to form a square.

The basis of this method is to discover a special value that when added to both sides of the quadratic that will create a perfect square trinomial.

That special value is found by evaluation the expression ${\left(\frac{b}{2}\right)}^{2}$ where $b$ is found in $a {x}^{2} + b x + c = 0$. Also in this explanation I assume that $a$ has a value of $1$.

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

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

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

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

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

A quadratic that is a perfect square is very easy to solve.

Please take a look at the video to see an example of completing the square visually.