IBMaths: AA SLDPRevision Notes2. Functions2.2 Quadratic Functions & Graphs2.2.4 Quadratic Inequalities

Quadratic Inequalities (DP IB Maths: AA SL)

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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
  • A common mistake is writing x minus h less than plus-or-minus square root of n or x minus h greater than plus-or-minus square root of n
    • This is NOT correct!
  • The correct solution to (x - h)2 < n is
    • open vertical bar x minus h close vertical bar less than square root of n which can be written as negative square root of n less than x minus h less than square root of n
    • The final solution is h minus square root of n less than x less than h plus square root of n
  • The correct solution to (x - h)2 > n is
    • open vertical bar x minus h close vertical bar greater than square root of n which can be written as x minus h less than negative square root of n or x minus h greater than square root of n
    • The final solution is x less than h minus square root of n or x greater than h plus square root of n

Exam Tip

  • It is easiest to sketch the graph of a quadratic when it has a positive  x squared  term, so rearrange first if necessary
  • Use your GDC to help select the correct region(s) for the inequality
  • Some makes/models of GDC may have the ability to solve inequalities directly
    • However unconventional notation may be used to display the answer (e.g. 6 space greater than space x space greater than space 3 rather than 3 space less than space x space less than space 6)
    • The safest method is to always sketch the graph

Worked example

Find the set of values which satisfy 3 x squared plus 2 x minus 6 greater than x squared plus 4 x minus 2.

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Dan

Author: Dan

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.