4F - Completing the square

Completing the square (CTS)

The process of completing the square (CTS) allows us to convert a quadratic in the general form (y = ax² + bx + c) into turning point form (y = a(x - h)² + k). In the turning point the turning point (vertex) is located at (h, k).

Method: Completing the square

Remember, to complete the square the coefficient of x² must be 1. If the coefficient is any number other than 1, simply divide all terms by the coefficient:

Once the coefficient of x² is 1, use the following steps to complete the square:

1. Halve the coefficient of x, then square it.
2. Add the value from (1) to the expression after the term involving x.
3. Subtract the same value to maintain the expression.
4. Factorise the first three terms as a perfect square.
5. Combine the remaining terms

The expression will now been in turning point form:

• Where the turning point occurs at (h, k).

4F - VIDEO EXAMPLE 1:

Factorise the following quadratic by completing the square:

• Therefore, the turning point occurs at (x, y) = (-3, -11).

4F - VIDEO EXAMPLE 2:

Factorise the following quadratic by completing the square:

• Therefore, the turning point occurs at (x, y) = (-4, 27).

4F - VIDEO EXAMPLE 3:

Factorise the following quadratic by completing the square:

• Therefore, the turning point occurs at (x, y) = (0.25, -4.25).

Solving quadratic equations by completing the square

Once a you have completed the square on a quadratic, it is possible to find the solutions to the equation by using inverse operations to isolate x.

• Remember: when you take the square root you will get a positive and negative solution

4F - VIDEO EXAMPLE 4:

Solve the following quadratic for x, given it is already in turning point form:

4F - VIDEO EXAMPLE 5:

Solve the following quadratic for x, by first completing the square:

The axis of symmetry

Completing the square on the general quadratic formula

Consider the general quadratic equation:

We can complete the square on the general form:

Therefore, after completing the square on the general quadratic formula the expression is:

The axis of symmetry

From the equation above we can see that the turning point (vertex) will exist upon the axis of symmetry defined by:

Where the values of a and b come from the general form: y= ax² + bx + c.

• The turning point of parabola lies on the axis of symmetry. Therefore, by determining the x-coordinate and substituting this into the function you can find the turning point without completing the square.

Figure 1 - The axis of symmetry lies half-way between the two x-intercepts.

4F - VIDEO EXAMPLE 6:

Use the axis of symmetry to determine the turning point for the following quadratic. Hence, determine the turning point form without completing the square.

Dynamic geometry representation of the axis of symmetry

The following GeoGebra gadget shows the axis of symmetry as the x-intercepts (m,0) and (n,0) are changed. As you can see the turning point (vertex) always exists on the line of symmetry for the parabola regardless of the value of the x-intercepts.