Mammoth Memory

Completing the square - example 4

Complete the square `x^2+7x+10=0`

Quick sketch


Complete the square and split the square into quarters using x square

 Fill the square in with the next term

Fill in

Fill in the table multiplying the y axis by the x axis


This is the same as `(x+7/2)^2`


If you add up each area you get:




Always plot this on a number line

Use the number line to work out the difference between the original number and the new number

The number line will help you remember

Original number `-`  New number


So     `x^2+7x+10=0`


Is the same as






Using a calculator


`x=-3.5+1.5`   and   `-3.5-1.5`

`x=-2`   and   `-5`


Now check



If `x=-2`              `(-2)^2+7times(-2)+10=0`

`+4-14+10=0`   Which is correct


If `x=-5`              `(-5)^2+7times(-5)+10=0`

`25-35+10=0`   Which is correct


The roots of `x^2+7x+10=0`   are `x=-2`   and  `x=-5`



This example has also been used in factorising quadratics (easy) and quadratic formula examples to show that the roots`-2`  and `-5`  can be found using any of these methods.