Edexcel A Level Maths: Pure:复习笔记2.4.2 Quadratic Inequalities

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

Can involve the discriminant or applications in mechanics and statistic

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
• • avoid multiplying or dividing by a negative numberif 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 positiveBe careful:

Solving quadratic inequalities on a calculator

• Be aware of unconventional ways calculators can display an answereg        8 > x > 2 rather than 2 < x < 8