# 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* $3.s.f.$

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