IB DP Maths: AA SL复习笔记2.2.4 Quadratic Inequalities

What affects the inequality sign when rearranging a quadratic inequality?

• The inequality sign is unchanged by...
  • Adding/subtracting a term to both sides
  • Multiplying/dividing both sides by a positive term
• The inequality sign flips (< changes to >) when...
  • Multiplying/dividing both sides by a negative term

How do I solve a quadratic inequality?

• STEP 1: Rearrange the inequality into quadratic form with a positive squared term
  • ax² + bx + c > 0
  • ax² + bx + c ≥ 0
  • ax² + bx + c < 0
  • ax² + bx + c ≤ 0
• 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 concave up so "U" shaped
• STEP 4: Identify the region that satisfies the inequality
  • If you want the graph to be above the x-axis then choose the region to be the two intervals outside of the two roots
  • If you want the graph to be below the x-axis then choose the region to be the interval between the two roots
  • For ax² + bx + c > 0
    • The solution is x < x₁ or x > x₂
  • For ax² + bx + c ≥ 0
    • The solution is x ≤ x₁ or x ≥ x₂
  • For ax² + bx + c < 0
    • The solution is x₁ < x < x­₂
  • For ax² + bx + c ≤ 0
    • The solution is x₁ ≤ x ≤ x­₂

How do I solve a quadratic inequality of the form (x - h)² < n or (x - h)² > n?