Express a quadratic in completed-square form, identify the turning point and line of symmetry, and use these to sketch the parabola

What this dot point is asking

SEAB wants you to take any quadratic written as $y = ax^2 + bx + c$ and rewrite it in completed-square form $y = a(x + p)^2 + q$. From that form you can read the turning point straight off, state the line of symmetry, decide whether the parabola has a minimum or a maximum, and produce a quick, correct sketch. This single technique underpins solving equations, finding ranges, and the maximum or minimum problems that appear all through the syllabus.

The answer

Why completed-square form is useful

A quadratic graph is a parabola. In the form $y = a(x + p)^2 + q$ every important feature is visible:

• the turning point is at $(-p,\ q)$,
• the line of symmetry is $x = -p$,
• the parabola has a minimum when $a > 0$ (opens upward) and a maximum when $a < 0$ (opens downward).

The reason is that $(x + p)^2$ is never negative. When $a > 0$ the smallest the whole expression can be is $q$, reached when $x = -p$; when $a < 0$ the largest it can be is $q$.

Completing the square when the coefficient of x squared is 1

For $x^2 + bx + c$, take half the coefficient of $x$, square it, add and subtract it:

The bracket $\left(x + \tfrac{b}{2}\right)^2$ captures the $x^2$ and $bx$ terms exactly; the $-\left(\tfrac{b}{2}\right)^2$ removes the extra constant the square introduced.

When the coefficient of x squared is not 1

If $a \ne 1$, factor $a$ out of the first two terms first, complete the square inside the bracket, then multiply back:

Keep the working tidy. The most common slip here is forgetting to multiply the constant you subtract by the factor $a$ when you expand the bracket back out.

Reading the graph

Once in completed-square form, sketch by plotting the turning point, drawing the axis of symmetry through it, and marking the $y$-intercept (the value of $y$ when $x = 0$, which is just $c$). Two or three points are enough for a clear sketch.

Examples in context

Example 1. Finding a minimum cost. Suppose a small business finds that its weekly cost in dollars is $C = x^2 - 10x + 40$, where $x$ is the number of items made in dozens. Completing the square gives $C = (x - 5)^2 + 15$, so the minimum cost is \$15 when$x = 5$. The completed-square form answers a real optimisation question with no calculus.

Example 2. Proving a quadratic is always positive. To show $x^2 + 2x + 5 > 0$ for every real $x$, complete the square: $x^2 + 2x + 5 = (x + 1)^2 + 4$. Because $(x + 1)^2 \ge 0$, the expression is at least $4$, so it is always positive. This is a clean way to prove a quadratic never touches the $x$-axis.

Try this

Q1. Express $x^2 + 8x + 3$ in the form $(x + p)^2 + q$. [2 marks]

• Cue. Half of $8$ is $4$, so $x^2 + 8x = (x + 4)^2 - 16$, giving $(x + 4)^2 - 13$.

Q2. Write down the turning point and line of symmetry of $y = (x - 3)^2 + 2$. [2 marks]

• Cue. Turning point $(3,\ 2)$; line of symmetry $x = 3$; it is a minimum since the coefficient of the squared bracket is positive.

Q3. Express $3x^2 - 12x + 5$ in completed-square form. [3 marks]

• Cue. $3(x^2 - 4x) + 5 = 3\big[(x - 2)^2 - 4\big] + 5 = 3(x - 2)^2 - 7$.