Algebra & Equations for Matric Maths

Key concepts

• Solving quadratic equations by factorising, formula, and completing the square
• Nature of roots — discriminant Δ = b² − 4ac
• Solving simultaneous equations (linear-linear and linear-quadratic)
• Quadratic and linear inequalities
• Equations with surds and exponents
• Rejecting invalid roots (from surd equations, zero denominators or context)
• Making one variable the subject before substituting

Worked examples

Solve for x: 2x² − 5x − 3 = 0

Factorise: (2x + 1)(x − 3) = 0. So x = −½ or x = 3. Always check both roots satisfy the original equation.

For which values of k does x² + kx + 4 = 0 have equal roots?

Equal roots ⇒ Δ = 0. k² − 4(1)(4) = 0, so k² = 16 and k = ±4.

Solve simultaneously: y = 2x − 1 and x² + y² = 13

Substitute the linear equation into the quadratic: x² + (2x − 1)² = 13, so 5x² − 4x − 12 = 0 and (5x + 6)(x − 2) = 0. If x = 2 then y = 3; if x = −6/5 then y = −17/5. Check both pairs in x² + y² = 13.

Solve x² − x − 6 < 0

Factorise: (x − 3)(x + 2) < 0. The critical values are −2 and 3. The parabola opens upwards, so it is negative between its roots: −2 < x < 3.

Solve √(x + 7) = x − 5

Square both sides: x + 7 = x² − 10x + 25, so x² − 11x + 18 = 0 and (x − 9)(x − 2) = 0. Check: x = 9 gives √16 = 4 = 9 − 5 ✓. x = 2 gives √9 = 3 but 2 − 5 = −3 ✗. So x = 9 only.

Common mistakes

• Dividing both sides by x (you lose the root x = 0).
• Forgetting ± when taking square roots.
• Flipping the inequality sign incorrectly when multiplying by a negative.
• Squaring both sides of a surd equation without checking for extraneous roots.
• Writing the solution of a quadratic inequality as two separate statements when it is one interval.
• Checking a simultaneous-equation solution in only one of the two equations.

Exam tips

• If the quadratic doesn't factorise quickly, jump to the formula — don't waste minutes.
• For inequalities, sketch the parabola and read the sign from the graph.
• State restrictions (e.g. x ≠ 0) before you start dividing.
• With one linear and one quadratic equation, make x or y the subject of the linear equation first.
• Give answers in the form asked for: exact, simplified or rounded to two decimal places.

In past papers

• NSC Nov 2024 Paper 1 Q1 — solving quadratics and inequalities
• NSC Nov 2023 Paper 1 Q1.4 — equations with surds

Frequently asked questions

What types of equations appear in Matric Paper 1?

Linear, quadratic, simultaneous, inequalities, surd equations and exponential/logarithmic equations. Question 1 of Paper 1 is almost always pure algebra.

Do I need to memorise the quadratic formula?

It is printed on the NSC information sheet, but you should still know it well: you need to identify a, b and c quickly, and the discriminant b² − 4ac inside it answers every nature-of-roots question.

How do I know when to reject a root?

Substitute each root into the original equation. Reject a root that makes a square root equal a negative number, makes a denominator zero, or does not fit the context, such as a negative length.

How do I solve a quadratic inequality quickly?

Make one side zero, factorise to find the critical values, then use a sign diagram or a rough sketch of the parabola to choose the interval where the expression has the required sign.