2.1 Completing the square

Solution by completing the square

We have seen that expressions of the form:

are known as differences of squares and can be factorised as follows:

This simple factorisation leads to another technique to solve quadratic equations known as completing the square .

We demonstrate with a simple example, by trying to solve for $x$ in:

We cannot easily find factors of this term, but the first two terms look similar to the first two terms of the perfect square:

However, we can cheat and create a perfect square by adding 2 to both sides of the equation in [link] as:

Now we know that:

which means that:

is a difference of squares. Therefore we can write:

The solution to $x^2-2x-1=0$ is then:

or

This means $x=1+\sqrt{2}$ or $x=1-\sqrt{2}$ . This example demonstrates the use of completing the square to solve a quadratic equation.

Method: solving quadratic equations by completing the square

1. Write the equation in the form $a{x}^2+bx+c=0$ . e.g. $x^2+2x-3=0$
2. Take the constant over to the right hand side of the equation. e.g. $x^2+2x=3$
3. If necessary, make the coefficient of the $x^2$ term = 1, by dividing through by the existing coefficient.
4. Take half the coefficient of the $x$ term, square it and add it to both sides of the equation. e.g. in $x^2+2x=3$ , half of the coefficient of the $x$ term is 1 and $1^2=1$ . Therefore we add 1 to both sides to get: $x^2+2x+1=3+1$ .
5. Write the left hand side as a perfect square: ${(x+1)}^2-4=0$
6. You should then be able to factorise the equation in terms of difference of squares and then solve for $x$ : $(x+1-2)(x+1+2)=0$

Solution by completing the square

Solve the following equations by completing the square:

1. $x^2+10x-2=0$
2. $x^2+4x+3=0$
3. $x^2+8x-5=0$
4. $2x^2+12x+4=0$
5. $x^2+5x+9=0$
6. $x^2+16x+10=0$
7. $3x^2+6x-2=0$
8. $z^2+8z-6=0$
9. $2z^2-11z=0$
10. $5+4z-z^2=0$