Solving by Completing the Square

Understand how to solve quadratic equations by completing the square using clear, step-by-step explanations, worked examples, and student-friendly notes.

1. Introduction

Completing the square is a method of rewriting a quadratic expression into a perfect square form. This makes the equation easier to solve. It works for every quadratic equation, even when factorisation is not possible.

The idea is to convert:

\(ax^2 + bx + c = 0\)

into a form like:

\((x + p)^2 = q\)

which can then be solved by taking square roots.

2. Understanding the Idea Behind Completing the Square

A quadratic expression like \(x^2 + bx\) can be made into a perfect square by adding:

\(\left(\dfrac{b}{2}\right)^2\)

Example:

\(x^2 + 6x → x^2 + 6x + 9 = (x + 3)^2\)

because:

\(\left(\dfrac{6}{2}\right)^2 = 9\)

3. Steps to Solve by Completing the Square

Use these steps to solve any quadratic equation:

  1. Write the equation in the form \(ax^2 + bx = -c\).
  2. If \(a \neq 1\), divide the entire equation by \(a\).
  3. Take half of the coefficient of \(x\), square it, and add to both sides.
  4. Rewrite the left side as a perfect square.
  5. Take square roots on both sides.
  6. Solve for \(x\).

3.1. When a = 1 (Simplest Case)

Example steps will be shown below.

3.2. When a ≠ 1 (Extra Step)

Divide all terms by \(a\) to make the coefficient of \(x^2\) equal to 1 before completing the square.

4. Worked Examples

These examples show how the method works in different situations:

4.1. Example 1: a = 1

Solve: \(x^2 + 4x - 5 = 0\)

  1. Move constant to other side: \(x^2 + 4x = 5\)
  2. Half of 4 = 2 → square = 4
  3. Add 4 to both sides: \(x^2 + 4x + 4 = 5 + 4\)
  4. Left side becomes a square: \((x + 2)^2 = 9\)
  5. Take square root: \(x + 2 = ±3\)
  6. Solutions: \(x = 1\) or \(x = -5\)

4.2. Example 2: a ≠ 1

Solve: \(2x^2 + 8x - 6 = 0\)

  1. Divide by 2: \(x^2 + 4x - 3 = 0\)
  2. Move constant: \(x^2 + 4x = 3\)
  3. Add \(\left(\dfrac{4}{2}\right)^2 = 4\): \(x^2 + 4x + 4 = 3 + 4\)
  4. Left side: \((x + 2)^2 = 7\)
  5. Square root: \(x + 2 = ±\sqrt{7}\)
  6. Solutions: \(x = -2 ± \sqrt{7}\)

4.3. Example 3: Fraction Steps

Solve: \(x^2 - x - 2 = 0\)

  1. Move constant: \(x^2 - x = 2\)
  2. Half of -1 = -1/2, square = 1/4
  3. Add 1/4 to both sides: \(x^2 - x + 1/4 = 2 + 1/4\)
  4. Left side: \((x - 1/2)^2 = 9/4\)
  5. Square root: \(x - 1/2 = ±3/2\)
  6. Solutions: \(x = 2\), \(x = -1\)

5. Geometric Meaning

Completing the square has a geometric interpretation: it represents adjusting the sides of a square to form a perfect square. This idea is used in the derivation of the quadratic formula.

6. Why Completing the Square Always Works

Unlike factorisation, which works only when the quadratic splits nicely, completing the square works for every quadratic equation. It systematically converts any quadratic expression to a perfect square form.

7. Common Mistakes

  • Forgetting to add the same value to both sides.
  • Incorrectly squaring half of the \(x\)-coefficient.
  • Sign errors when moving terms.
  • Taking square roots incorrectly (forgetting the ± sign).
  • Not dividing by \(a\) when \(a ≠ 1\).

8. Quick Practice

Solve the following by completing the square:

  1. \(x^2 + 6x + 2 = 0\)
  2. \(2x^2 - 4x - 3 = 0\)
  3. \(x^2 - 10x + 4 = 0\)
  4. \(3x^2 + 9x = 12\)

9. Summary

  • Completing the square rewrites the equation in the form \((x + p)^2 = q\).
  • Add \(\left(\dfrac{b}{2a}\right)^2\) after dividing by \(a\).
  • Take square roots and solve for \(x\).
  • This method works for all quadratic equations.