Completing the square - example 4

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

Quick sketch

Remember

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

NOTE:

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

If you add up each area you get:

`x^2+7/2x+7/2x+7/2times7/2`

`x^2+7x+49/4`

`x^2+7x+12\1/4`

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

`10-12\1/4=-2\1/4`

So `x^2+7x+10=0`

Is the same as

`(x+7/2)^2-2\1/4=0`

`(x+7/2)^2=2\1/4`

`x+7/2=+-sqrt(2\1/4)`

`x=-7/2+-sqrt(2\1/4)`

Using a calculator

`x=-3.5+-1.5`

`x=-3.5+1.5` and `-3.5-1.5`

`x=-2` and `-5`

Now check

`x^2+7x+10=0`

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

Answer:

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

NOTE:

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.