CIE A Level Maths: Pure 1复习笔记1.2.5 Quadratic Inequalities
Quadratic Inequalities
Quadratic inequalities
- Similar to quadratic equations
- Sketching a quadratic graph is essential
- Can involve the?discriminant?or?applications?in?mechanics?and?statistics
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
- For?ax2?+?bx?+?c?> 0?you want the region?above?the?x-axis
- Be careful:
- 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 positive
Worked Example
Exam Tip
- A calculator can be super-efficient but some marks are for method.
- Use your judgement:
- is it a “show that” or “prove” question?
- how many marks?
- how long is the question?
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