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The Quadratic Formula

When factorising is hard, one formula solves any quadratic equation.

The Formula

Not every quadratic factorises easily with whole numbers. The quadratic formula solves any equation written as ax² + bx + c = 0 .

The \pm symbol means "plus or minus" — it produces two answers from one formula: one using +, one using -. Always identify a, b, and c first, matching your equation to ax² + bx + c = 0 exactly, including signs.

Example 1 — checking against factorising

Solve x^{2} + 5 x + 6 = 0 using the formula. 1 Identify: a = 1, b = 5, c = 6 2 Work out b^{2} - 4 ac: 5^{2} - 4 (1)(6) = 25 - 24 = 1 3 Square root it: \sqrt 1 = 1 4 x = (- 5 \pm 1) / (2 \times 1) = (- 5 \pm 1) / 2 5 Two answers: x = (- 5 + 1) \div 2 = - 2, or x = (- 5 - 1) \div 2 = - 3 Answer: x = - 2 or x = - 3 (matches the factorising method from the last lesson)

Example 2 — a case that does not factorise easily

Solve x^{2} + 4 x + 1 = 0 . 1 Identify: a = 1, b = 4, c = 1 2 Work out b^{2} - 4 ac: 4^{2} - 4 (1)(1) = 16 - 4 = 12 3 Square root it: \sqrt 12 \approx 3.46 4 x = (- 4 \pm 3.46) / 2 5 Two answers: x \approx (- 4 + 3.46) \div 2 \approx - 0.27, or x \approx (- 4 - 3.46) \div 2 \approx - 3.73 Answer: x \approx - 0.27 or x \approx - 3.73

Example 3 — a leading coefficient

Solve 2 x^{2} - 3 x - 5 = 0 . 1 Identify: a = 2, b = - 3, c = - 5 2 Work out b^{2} - 4 ac: (- 3) ² - 4 (2) (- 5) = 9 + 40 = 49 3 Square root it: \sqrt 49 = 7 4 x = (3 \pm 7) / (2 \times 2) = (3 \pm 7) / 4 5 Two answers: x = \frac{10}{4} = 2.5, or x = - \frac{4}{4} = - 1 Answer: x = 2.5 or x = - 1

What the discriminant tells you
What the discriminant tells you

Practice

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For 3 x^{2} + 2 x - 8 = 0, what are a, b, and c?

For x^{2} - 2 x - 15 = 0, work out the discriminant b^{2} - 4 ac.

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