Algebraic Techniques (HSC SSCE Mathematics Advanced): Revision Notes

Completing the Square

Key idea

Completing the square is a technique that transforms a quadratic expression from its standard form into vertex form. This transformation reveals important features of the parabola and makes certain calculations much easier.

Standard form: $ax^2 + bx + c$

Vertex form: $a(x - h)^2 + k$

The process involves creating a perfect square trinomial by strategically adding and subtracting the same value, keeping the expression mathematically equivalent to the original.

What vertex form tells you

Once a quadratic is written in vertex form $a(x - h)^2 + k$, you can immediately identify several key features of the parabola:

From the vertex form $a(x - h)^2 + k$, you can determine:

• Turning point (vertex): The coordinates are $(h, k)$
• Axis of symmetry: The vertical line $x = h$
• Direction of opening: If $a > 0$, the parabola opens upward; if $a < 0$, it opens downward

Understanding the sign of h

A common source of confusion is reading the value of $h$ from the vertex form. Pay careful attention to the sign inside the brackets:

• $(x - h)^2$ means $h$ is positive
• $(x + h)^2$ means $h$ is negative

Method when a = 1

When the coefficient of $x^2$ is $1$, follow these five steps to complete the square:

Step	What to do
1	Take the first two terms: $x^2 + bx$
2	Halve the coefficient of $x$, then square it: $\left(\frac{b}{2}\right)^2$
3	Add and subtract $\left(\frac{b}{2}\right)^2$ after the first two terms
4	Factorise the first three terms into a perfect square
5	Simplify the remaining constant

Worked example 1

From this vertex form, we can see that the turning point is at $(-3, -4)$ and the axis of symmetry is $x = -3$.

Method when a ≠ 1

When the coefficient of $x^2$ is not equal to $1$, you must factor out the value of a from the first two terms before completing the square.

Worked example 2

Worked example 3 — finding the turning point

The graph shows the parabola with its vertex at $(2, -3)$ and the axis of symmetry at $x = 2$.

Common mistakes

Exam tips

When completing the square in an examination:

• Always write the formula explicitly: Show $(b \div 2)^2$ clearly in your working for full marks
• Show both the addition and subtraction: Write both $+ \left(\frac{b}{2}\right)^2$ and $- \left(\frac{b}{2}\right)^2$ clearly in your working
• State the turning point as coordinates: Write it as an ordered pair $(h, k)$
• For a ≠ 1 questions: Factor out a, complete the square, then expand a back through the expression
• Double-check the sign of h: Before writing your final answer, verify that you've read $h$ correctly from the vertex form

Remember!