Completing the square

We have learned how to factorise a quadratic, but there are some quadratic equations that cannot be solved by factorising, and so we need to learn how to complete the square.

This involves taking a quadratic in the form $a x^{2} + b x + c$ and getting it into the form

p (x - q)^{2} + r

Doing this allows us to solve any quadratic equation, not just ones that can be easily factorised.

Example

Express each quadratic below in complete square form, and hence find the roots of the quadratic.

$x^{2} + 4 x + 3$
$x^{2} - 6 x + 10$
$x^{2} + 3 x + 5$

Exercise

Hint

Solution

Completing the square ​

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Completing the square

Example

Exercise