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Quadratic equations and inequalities Cambridge IGCSE Additional Mathematics revision

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In plain words

Solving a quadratic equation gives points. Solving a quadratic inequality gives a stretch of the number line, or two stretches. A quick sketch of the parabola shows which.

5 things to know

  1. A quadratic equation can be solved by factorising, by the quadratic formula, or by completing the square.
  2. To solve a quadratic inequality: make one side 0, find the roots (the critical values), sketch the parabola, and read off where it is above or below the x-axis.
  3. For a U-shaped parabola with roots a and b, where a < b: it is below the axis between the roots, and above the axis outside them.
  4. So (x − a)(x − b) < 0 gives a < x < b, and (x − a)(x − b) > 0 gives x < a or x > b.
  5. Write "between" answers as one statement, −3 < x < 4. Write "outside" answers as two, x < 1 or x > 6.

Worked example

Solve x² − x − 12 < 0.

  1. Factorise: (x − 4)(x + 3) < 0. The critical values are −3 and 4.
  2. The parabola is U-shaped, so it is below the x-axis between its roots.
  3. −3 < x < 4.

Worked example

Solve x² − 7x + 6 > 0.

  1. Factorise: (x − 1)(x − 6) > 0. The critical values are 1 and 6.
  2. The parabola is above the x-axis outside its roots.
  3. x < 1 or x > 6.

Tips and tricks

  • Never write "1 > x > 6" for the outside case. No number is both less than 1 and greater than 6. Use the word "or".
  • If the x² term is negative, multiply through by −1 first and turn the inequality sign round.
5 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

Quadratic equations and inequalities: 5 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

  1. What is the solution of (x − 2)(x − 5) < 0?
    • x < 2 or x > 5
    • 2 < x < 5 (the answer)
    • x < 2
    • x > 5

    The parabola is below the axis between its roots.

  2. What is the solution of x² > 9?
    • x > 3
    • −3 < x < 3
    • x < −3 or x > 3 (the answer)
    • x > ±3

    Outside the roots −3 and 3.

  3. What are the roots of x² − 4x − 5 = 0?
    • 1 and −5
    • −1 and 5 (the answer)
    • 1 and 5
    • −1 and −5

    (x − 5)(x + 1) = 0.

  4. What is the solution of x² + 2x − 8 ≥ 0?
    • −4 ≤ x ≤ 2
    • x ≤ −4 or x ≥ 2 (the answer)
    • x ≥ 2
    • −2 ≤ x ≤ 4

    (x + 4)(x − 2) ≥ 0: outside the roots, including them.

  5. What are the roots of 2x² − 3x − 2 = 0?
    • 2 and 1/2
    • 2 and −1/2 (the answer)
    • −2 and 1/2
    • −2 and −1/2

    (2x + 1)(x − 2) = 0.

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