IB DP Maths: AA HL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?
- The safest way is by following the steps above
- Expand and rearrange
Exam Tip
Worked Example
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