Complete the square

In a nutshell

Completing the square is a method that can be used to solve quadratic equations. It involves rearranging the quadratic into a form which can make it easier to solve. It is another way of solving a quadratic equation, instead of factorising or using the quadratic formula.

Complete the square

To complete the square on a quadratic expression, it should be put into the form

$(x+a)^2+b$

Once the quadratic has been put in the completed square form, it can be rearranged to solve for $x$ .

PROCEDURE

$1.$	Write the quadratic expression in descending powers of $x$ .
$2.$	Set up the answer in the form $(x \qquad )^2$ .
$3$ .	Take the co-efficient of $x$ from the quadratic, half the number and fill this number in the bracket.
$4.$	Take the number that has just been filled in the bracket, square the number and subtract this number from the bracket.
$5.$	Add the constant term.

Example 1

Complete the square on the quadratic expression

$x^2+6x$

Start by setting up the answer in the form

$(x \qquad)^2$

Take the co-efficient of $x$ , in this case $6$ and half the number, which gives $3$ . Fill this number in the bracket.

$(x+3)^2$

Take the number in the bracket, $3$ , and square it. Subtract this number from the bracket. This gives

$\underline{(x+3)^2-9}$

Example 2

Complete the square on the quadratic expression

$x^2+10x+6$

Start by setting up the answer in the form

$(x \qquad)^2$

Take the co-efficient of $x$ , in this case $10$ and half the number, which gives $5$ . Fill this number in the bracket.

$(x+5)^2$

Take the number in the bracket, $5$ , and square it. Subtract this number from the bracket and add the constant term. This gives

$(x+5)^2-25+6 \\ \underline{(x+5)^2 - 19}$

Solve a quadratic by completing the square

Once a quadratic has been rearranged into the completed square form, it is possible to solve the equation.

Example 3

Solve

$x^2+10x+6=0$

This has already been rearranged into the correct form in example 2 above.

$(x+5)^2-19=0$

To solve the quadratic, take $19$ to the other side of the equation and then square root, before taking $5$ over.

$\begin {aligned}(x+5)^2-19&=0 \\(x+5)^2 &= 19 \\x+5 &= \pm \sqrt {19} \\x &= -5 \pm \sqrt {19}\end {aligned}$

The answer can be left in surd form, or can be calculated to $3.s.f.$

$\underline{x= -9.36 \space or \space x=-0.641}$