Grade 11 Equations and Inequalities

Grade 11 equations and inequalities: quadratic formula, completing the square, nature of roots, quadratic inequalities and simultaneous equations, with…

Algebraic expressions, equations and inequalities are the largest block of Grade 11 Paper 1, at about 45 of the 150 marks. The focus is the quadratic equation: you solve it by factorising, by completing the square and with the quadratic formula, and you use the discriminant to describe the nature of the roots without solving.

Grade 11 also adds quadratic inequalities, simultaneous equations where one equation is linear and the other quadratic, and equations that reduce to quadratics with a substitution, such as x⁴ − 5x² + 4 = 0. These skills feed straight into Grade 11 functions (intersections of graphs) and Grade 12 calculus.

Key concepts

  • Quadratic formula: x = [−b ± √(b² − 4ac)] / 2a
  • Completing the square to write ax² + bx + c in the form a(x + p)² + q
  • Nature of roots from Δ = b² − 4ac: real/non-real, equal/unequal, rational/irrational
  • Quadratic inequalities solved with a sketch or a sign table
  • Simultaneous equations: one linear and one quadratic
  • Equations reducible to quadratics with a substitution, e.g. k = x²

Worked examples

Solve, correct to two decimal places: 2x² − 3x − 4 = 0

a = 2, b = −3, c = −4. x = [3 ± √(9 + 32)] / 4 = [3 ± √41] / 4, so x ≈ 2.35 or x ≈ −0.85.

For which values of k does x² − 6x + k = 0 have two real, unequal roots?

Δ = 36 − 4k. Two real, unequal roots need Δ > 0: 36 − 4k > 0, so k < 9.

Solve: x² − x − 12 < 0

Factorise: (x − 4)(x + 3) < 0. The critical values are −3 and 4. The parabola opens upwards, so it is negative between its roots: −3 < x < 4.

Solve simultaneously: y = x + 1 and y = x² − 1

Set the right-hand sides equal: x² − 1 = x + 1, so x² − x − 2 = 0 and (x − 2)(x + 1) = 0. If x = 2, y = 3; if x = −1, y = 0. The solutions are (2; 3) and (−1; 0).

Common mistakes

  • Dropping the minus sign in −b when b is negative.
  • Dividing only part of the numerator by 2a in the quadratic formula.
  • Writing a quadratic inequality answer as two separate inequalities when it is one interval.
  • Forgetting to find the y-values in simultaneous equations.

Exam tips

  • Write down a, b and c before substituting into the formula.
  • If a question says 'correct to two decimal places', it expects the quadratic formula.
  • For nature of roots, state the condition (Δ > 0, Δ = 0 or Δ < 0) before solving for k.

Frequently asked questions

When should I use the quadratic formula instead of factorising?

Try factorising first for about 20 seconds. If it doesn't factorise neatly, or the question asks for decimal answers, use the formula.

What does the discriminant tell me?

Δ > 0 gives two real, unequal roots (rational if Δ is a perfect square); Δ = 0 gives equal roots; Δ < 0 gives no real roots.

How many marks is algebra in Grade 11?

Algebraic expressions, equations and inequalities are about 45 of the 150 marks in Paper 1, according to the CAPS examination guidelines.

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