Completing the Square: Concepts, Formulas and Examples

Completing the square is a technique that rewrites a quadratic ax squared + bx + c in the vertex form a(x - h) squared + k. It works by adding and subtracting the right constant so that part of the expression becomes a perfect square trinomial, which then collapses into a squared bracket.

This matters for three reasons: it solves any quadratic equation, it reveals the vertex (h, k) of the parabola directly, and it is the exact argument used to derive the quadratic formula. One method, three payoffs.

Key Concepts

• Perfect square trinomial: x squared + 2mx + m squared = (x + m) squared. Recognising this pattern is the heart of the method.
• Vertex form: a(x - h) squared + k, where (h, k) is the vertex of the parabola, its maximum or minimum point.
• The magic number: for x squared + bx, add (b/2) squared to complete the square. Half the x-coefficient, then square it.
• Balance: whatever you add, you must also subtract (or add inside the equation consistently) so the value does not change.

How It Works

1. If a is not 1, divide the whole equation by a first so the x squared coefficient becomes 1.
2. Move the constant term to the right side of the equation.
3. Take half of the x-coefficient, square it, and add it to both sides.
4. The left side is now a perfect square: write it as (x + b/2) squared.
5. Take square roots of both sides, remembering the plus-or-minus, and solve for x.

Worked Example

Solve x squared + 6x + 5 = 0 by completing the square.

1. The x squared coefficient is already 1. Move the constant: x squared + 6x = -5.
2. Half of 6 is 3, and 3 squared is 9. Add 9 to both sides: x squared + 6x + 9 = -5 + 9.
3. Left side collapses: (x + 3) squared = 4.
4. Square root both sides: x + 3 = plus or minus 2.
5. So x = -3 + 2 = -1 or x = -3 - 2 = -5.

Check x = -1: 1 - 6 + 5 = 0. Check x = -5: 25 - 30 + 5 = 0. Both verified.

Common Misconceptions

• You add (b/2) squared to only one side. Wrong. Adding it to only one side changes the equation. Add it to both sides, or add and subtract it on the same side.
• The square root of 4 gives only x + 3 = 2. Wrong. (x + 3) squared = 4 means x + 3 = 2 or x + 3 = -2. Forgetting the negative root loses half the solutions.
• The method only works when a = 1. Wrong. Divide through by a first; for 2x squared + 8x + 6 = 0, divide by 2 to get x squared + 4x + 3 = 0, then proceed.

Real-World Applications

• In optimisation, completing the square finds the maximum or minimum of a quadratic function, such as the peak of a projectile or the best price for maximum profit.
• In geometry, it converts the general equation of a circle into centre-radius form.
• In physics, energy equations are often completed to squares to identify minimum-energy configurations.

To recap: halve the x-coefficient, square it, add it to both sides, collapse into a squared bracket, then take plus-or-minus square roots. The same steps also prove the quadratic formula, so mastering them pays off twice.