### GCSE Maths

Numbers
Algebra
Geometry and Measures
Probability
Statistics

# Completing the Square

Completing the square is an algebraic technique used to solve quadratic equations and rewrite them in a more convenient form. This method involves transforming the quadratic equation from the standard form: into the form: • It’s important to remember that and In doing so, we are effectively ‘completing’ the square of the quadratic expression by adding and subtracting a specific value to make it a perfect square trinomial.

A perfect square trinomial is a quadratic expression that can be factored into the product of two identical binomials. For example: This quadratic expression can be factored into the product of two identical binomials: Simplifying this expression, we get: So, is a perfect square trinomial that can be factored into .

Completing the square is very helpful when factoring or using the quadratic formula isn’t easily applicable. It can also be used to find the highest or lowest point of a curve represented by the quadratic equation.

## Solving Quadratic Equations by Completing the Square

Before we look at a worked example, let’s look through the steps for completing the square:

Step 1: Write the quadratic equation in standard form .

Step 2: If the coefficient of x^2 (a) is not 1, factor it out from the quadratic terms.

Step 3: Divide the coefficient of x (b) by 2, square the result, and add and subtract it within parentheses.

Step 4: Combine the terms in the parentheses to form a binomial squared and rewrite the remaining constant outside the parentheses as e.

Now, let’s look at a worked example:

Solve the quadratic equation Our goal is to rewrite this equation in the form of (x – d)^2 + e = 0, where d and e are constants. To achieve this, we’ll complete the square. Here are the steps:

1. Identify the coefficient of the linear term, which is -6 in this case.

• As the coefficient is negative, we are aiming to rewrite the equation in the form of and not .

2. Take half of this coefficient: .

3. Square the result from step 2: .

4. Add and subtract the result from step 3 to the left side of the equation: 5. Factor the first three terms into a perfect square: Now, the equation is in the form of , where d = 3 and e = 4.

To solve for x, you can either use the square root property or the quadratic formula.

Using the square root property:   So, the solutions are   Therefore, the solutions for the quadratic equation are and .

The process is slightly different when the coefficient of the term is not 1. This is because you have to factor the coefficient a of the term out of the quadratic and linear terms, leaving you with .

## Examples

Example 1:

Solve the equation Start by grouping the quadratic and linear terms: Complete the square by adding and subtracting the square of half of the linear coefficient: Rewrite as a binomial squared plus a constant: Now, isolate x:   Example 2:

Solve the equation Factor out the coefficient of : Complete the square: Rewrite as a binomial squared plus a constant: Isolate :    Example 3:

Write the expression in the form , where a, b and c are constants.

1. Factor the coefficient of x² (which is 2) out of the quadratic and linear terms: 2. Complete the square for the expression inside the parentheses:

To complete the square, we need to find a value that can be added and subtracted inside the parentheses. We’ll take half of the coefficient of x, which is 4, and square it: (4/2)² = 2² = 4.

Now we add and subtract this value inside the parentheses: 3. Rewrite the quadratic expression inside the parentheses as a square of a binomial: 4. Distribute the coefficient of x² (2) back in: 5. Simplify the constant term: 