Solving Quadratic Inequalities (Edexcel GCSE Maths)

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Solving Quadratic Inequalities

What are quadratic inequalities?

  • Similar to quadratic equations quadratic inequalities just mean there is a range of values that satisfy the solution
  • Sketching a quadratic graph is essential

2.4.2 Quadratic Inequalities Notes Diagram 1, Edexcel A Level Maths: Pure revision notes 

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 + = 0 to get x1 and xwhere 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 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
    • 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. Be careful:

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

Examiner Tip

  • Always start by rearranging to a quadratic with positive squared term
  • Always sketch a graph of the quadratic before deciding the final answer

Worked example

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Daniel I

Author: Daniel I

Expertise: Maths

Daniel has taught maths for over 10 years in a variety of settings, covering GCSE, IGCSE, A-level and IB. The more he taught maths, the more he appreciated its beauty. He loves breaking tricky topics down into a way they can be easily understood by students, and creating resources that help to do this.