Quadratic Inequalities

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

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 x₂where 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

2.4.2 Quadratic Inequalities Notes Diagram 3, Edexcel A Level Maths: Pure revision notes

1-2-5-quadratic-inequalities-example-diagram

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?

Quadratic Inequalities (CIE A Level Maths: Pure 1)