Quadratic Equations for Matric Maths

Factorising, quadratic formula, completing the square and nature of roots for NSC Paper 1. AI step-by-step explanations.

Quadratic equations are tested in every Matric Paper 1. You'll need to solve by factorising, by the quadratic formula and (for nature of roots) by analysing the discriminant. Choose the method from the question: factorise when the factors are clear, use the formula when they are not, and show the discriminant separately whenever the question asks about the number or type of roots. Quadratics also appear inside other questions: finding the x-intercepts of a parabola, finding n in a quadratic pattern, or finding where a line meets a curve.

Key concepts

  • Factorising trinomials
  • Quadratic formula x = (−b ± √(b² − 4ac)) / 2a
  • Completing the square
  • Discriminant Δ = b² − 4ac and nature of roots
  • Writing the equation in standard form ax² + bx + c = 0 first
  • Values of k for real, equal or non-real roots
  • Quadratics in disguise, e.g. x⁴ − 5x² + 4 = 0

Worked examples

Solve x² − 6x + 5 = 0.

(x − 1)(x − 5) = 0 ⇒ x = 1 or x = 5.

Solve x(x − 4) = 12

Expand and rearrange: x² − 4x − 12 = 0, so (x − 6)(x + 2) = 0 and x = 6 or x = −2.

Solve 2x² + 3x − 7 = 0 correct to two decimal places.

a = 2, b = 3, c = −7. x = (−3 ± √(9 + 56)) / 4 = (−3 ± √65) / 4, so x ≈ 1.27 or x ≈ −2.77.

For which values of k does x² − 4x + k = 0 have non-real roots?

Non-real roots need Δ < 0: (−4)² − 4(1)(k) < 0, so 16 − 4k < 0 and k > 4.

Solve x⁴ − 5x² + 4 = 0.

Let y = x²: y² − 5y + 4 = 0, so (y − 1)(y − 4) = 0 and y = 1 or y = 4. Then x² = 1 or x² = 4, so x = ±1 or x = ±2.

Solve x² + 6x − 2 = 0 by completing the square.

x² + 6x = 2, so x² + 6x + 9 = 11 and (x + 3)² = 11. Therefore x = −3 ± √11.

Common mistakes

  • Sign errors in the formula — write b, a, c clearly before substituting.
  • Forgetting that 'real and equal' means Δ = 0, not Δ > 0.
  • Factorising before the equation is in standard form (= 0).
  • Forgetting to substitute back after a substitution such as y = x².

Exam tips

  • If a quadratic doesn't factorise within 20 seconds, switch to the formula.
  • For nature of roots, write Δ = b² − 4ac with the values substituted before simplifying.
  • Keep the surd form until the last line, then round to the number of decimals asked for.

In past papers

  • NSC Nov 2024 Paper 1 Q1.1–1.3

Frequently asked questions

When should I use completing the square?

Whenever you need the turning point of a parabola or to derive the quadratic formula.

What does the discriminant tell me?

If Δ > 0 the roots are real and unequal (and rational when Δ is a perfect square, for rational coefficients). If Δ = 0 they are real and equal, and if Δ < 0 they are non-real.

Can I always use the quadratic formula?

Yes, it works for every quadratic equation. Factorising is quicker when the factors are obvious, and some questions specifically ask you to factorise or complete the square.

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