GradeGen.AI

What is Completing the Square?

When sketching a curved graph or finding its exact lowest point, the standard $x^2 + bx + c$ format is not always helpful.

Completing the square is an algebraic technique used to rewrite a standard quadratic expression into a format known as vertex form.
The vertex form looks like $(x + p)^2 + q$ or .

The Method for Monic Quadratics

A monic quadratic is an expression where the coefficient (the number in front) of $x^2$ is exactly 1, written as $x^2 + bx + c$ .
To transform this, you must apply the "halving" rule to find $p$ : simply halve the coefficient of $x$ .
However, expanding $(x + p)^2$ generates an extra, unwanted constant ( $p^2$ ).
To maintain equality, you must immediately apply the "subtraction" rule: always subtract $p^2$ outside the bracket before adding the original constant $c$ .

x^2 + bx + c = \left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2 + c

Worked Example: Monic Quadratic

Calculate the completed square form of the quadratic expression $x^2 + 6x - 2$ .

Step 1: Halve the coefficient of $x$ to create the perfect square bracket.

Half of $6$ is $3$ .
Write the bracket: $(x + 3)^2$

Step 2: Subtract the square of this halved value outside the bracket.

$3^2 = 9$
Expression becomes: $(x + 3)^2 - 9$

Step 3: Bring down the original constant and simplify to reach the final answer.

Add the original $- 2$ .
$(x + 3)^2 - 9 - 2$
Final answer: $(x + 3)^2 - 11$

Worked Example: Monic Quadratic with an Odd Coefficient

Calculate the completed square form of the quadratic expression $x^2 - 5x + 4$ .

Step 1: Halve the coefficient of $x$ .

Half of $-5$ is $-\frac{5}{2}$ . OCR exams prefer exact fractions over decimals.
Write the bracket: $\left(x - \frac{5}{2}\right)^2$

Step 2: Subtract the square of this fraction.

$\left(-\frac{5}{2}\right)^2 = \frac{25}{4}$

Step 3: Bring down the original constant, find a common denominator, and simplify.

Add the original $+ 4$ , which is equivalent to $\frac{16}{4}$ .
$\left(x - \frac{5}{2}\right)^2 - \frac{25}{4} + \frac{16}{4}$

The Method for Non-Monic Quadratics

But what happens if you have $3x^2$ or $-2x^2$ instead of just a single $x^2$ ?

A non-monic expression is a quadratic where the leading coefficient $a$ is any value other than 1.
You must factorise this value of $a$ out of the $x^2$ and $x$ terms before completing the square.
Use large square brackets $[ ]$ to separate the internal completing the square process from the external multiplier.

Worked Example: Non-Monic Quadratic

Calculate the completed square form of the expression $2x^2 - 7x + 1$ .

Step 1: Factorise the leading coefficient ( $2$ ) out of the $x^2$ and $x$ terms.

$2\left[x^2 - \frac{7}{2}x\right] + 1$

Step 2: Complete the square inside the square brackets by halving the new $x$ -coefficient.

Half of $-\frac{7}{2}$ is $-\frac{7}{4}$ .

Step 3: Square the internal fraction.

$2\left[\left(x - \frac{7}{4}\right)^2 - \frac{49}{16}\right] + 1$

Step 4: Multiply the external factor back into the square bracket and simplify the constants.

Multiply both terms inside by $2$ : $2\left(x - \frac{7}{4}\right)^2 - \frac{98}{16} + 1$

Finding the Turning Point

The vertex form is not just a neat algebraic trick; it instantly reveals the exact coordinates where a parabola changes direction.

For any quadratic in the form $y = a(x + p)^2 + q$ , the turning point (vertex) is at $(-p, q)$ .

Geometric Transformations

Completing the square also acts as a set of instructions for transforming the parent graph $y = x^2$ .
The value $-p$ shifts the graph horizontally, and $q$ shifts it vertically. In OCR exams, this translation is often written as the column vector $(\begin{array}{c} - p \end{array}$ .

Students often incorrectly write $(x + b)^2$ instead of halving the coefficient to get $(x + \frac{b}{2})^2$ .

When extracting the turning point from $(x + p)^2 + q$ , remember to flip the sign of the value inside the bracket so the $x$ -coordinate is $-p$ .

For non-monic quadratics, always remember to multiply the subtracted squared term by the external factor $a$ before you simplify the final constant.

If an OCR question says 'Hence, write down the coordinates of the turning point...', you must extract the coordinates directly from your completed square form in the previous part to secure the marks.

An algebraic method used to rewrite a quadratic expression as a squared bracket plus or minus a constant.

A mathematical phrase containing a variable squared as its highest power, typically taking the form $ax^2 + bx + c$ .

The structural form of a quadratic expression written as $a(x + p)^2 + q$ , which reveals the highest or lowest point of its graph.

An algebraic expression that can be factored into identical binomials, written as a single bracket squared, such as $(x + p)^2$ .

A numerical or constant quantity placed before and multiplying the variable in an algebraic expression.

A polynomial where the coefficient of the highest-degree term (the $x^2$ term in quadratics) is exactly 1.

A quadratic expression where the leading coefficient $a$ is any value other than 1.

The specific coordinate on a curve where the gradient becomes zero and the graph changes direction.

The lowest point on a U-shaped parabola, occurring when the $x^2$ coefficient is positive.

The highest point on an n-shaped parabola, occurring when the $x^2$ coefficient is negative.

A specific U-shaped or n-shaped curve that represents the graph of a quadratic function.

A mathematical notation $\begin{pmatrix} x \\ y \end{pmatrix}$ used to describe translations or displacements.