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?<?x-2
- For?ax2?+?bx?+?c ≤ 0
- The solution is?x1?≤?x?≤?x-2
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
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