Quadratic Inequalities (Cambridge International A-Level Mathematics)

Quadratic inequalities

What is a quadratic inequality?

• Similar to quadratic equations
• Sketching a quadratic graph is essential
• Can involve the discriminant or applications in mechanics and statistics

How do I solve quadratic inequalities?

• STEP 1: Rearrange the inequality into quadratic form with a positive squared term

  • ax²+ bx + c > 0 (>, <, ≤ or ≥)

• STEP 2: Find the roots of the quadratic equation

  • Solve ax² + bx + c = 0 to get x₁* and x₂ where x₁ < x₂

• STEP 3: Sketch a graph of the quadratic and label the roots

  • As the squared term is positive it will be "U" shaped

• STEP 4: Identify the region that satisfies the inequality

  • For ax²+ bx + c > 0 you want the region above the x-axis

    • The solution is x< x₁ or **x > x₂ **

  • For ax²** + bx + c < 0** you want the region below the x-axis

    • The solution is x > x₁ and **x < x₂
    • This is more commonly written as x₁** < x < x₂

• Be careful:

  • avoid multiplying or dividing by a negative number if unavoidable, “flip” the inequality sign so < → >, ≥ → ≤, etc
  • avoid multiplying or dividing by a variable (x) that could be negative (multiplying or dividing by x² guarantees positivity (unless x could be 0) but this can create extra, invalid solutions)
  • do rearrange to make the x² term positive

Worked Example

Exam Hint

• A calculator can be super-efficient but some marks are for method.

• Use your judgement:

  • is it a “show that” or “prove” question?
  • how many marks?
  • how long is the question?