Completing the Square

Understanding the Concept

Completing the square is a method used to solve quadratic equations, rewrite quadratic functions, and evaluate integrals.
The key concept in completing the square is to rewrite the quadratic expression in the form (x - h)² + k.

Recognise the Structure

A quadratic equation is typically written in the form ax² + bx + c = 0.
When a quadratic equation is expressed in the form (x - h)² + k = 0, with h and k as constants, it has been ‘completed the square’.

The Process

To complete the square, follow these steps:
- First, the coefficient of x² should be 1. If it’s not, then divide the whole equation by the coefficient.
- Rewrite the quadratic as (x ± b/2)² = c. Here, (x ± b/2) is the linear term divided by 2 and squared, and c is a constant.
- Look at the number in the x term (the b value), halve it, square it, and then add it inside the brackets but subtract it on the outside again in order to maintain the balance of the equation.
After completing the square, the equation can be solved through further simplification.

Examples

To complete the square for the equation x² + 6x + 8 = 0, write (x + 3)², but because (x + 3)² = x² + 6x + 9 and you want x² + 6x + 8, you must also subtract 1. So, this quadratic equation as a completed square would be (x + 3)² - 1 = 0.
To solve this equation, (x + 3)² - 1 = 0, for x, rearrange to (x + 3)² = 1. So x + 3 could be either 1 or -1. Hence the solutions are x = -2 or x = -4.

Practice

Completing the square is an important step in solving quadratic equations, converting to vertex form, and graphing quadratic functions. It’s crucial to understand this method and practice it on a variety of quadratic equations.

Remember, identifying the quadratic structure, accurately halving and squaring the linear coefficient, and maintaining the balance of the equation are pivotal to mastering completing the square.