Quadratic inequalities

• Similar to quadratic equations quadratic inequalities just mean there is a range of values that satisfy the solution
• Sketching a quadratic graph is essential

Can involve the?discriminant?or?applications?in?mechanics?and?statistic

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
• • 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 positiveBe careful:

Solving quadratic inequalities on a calculator

• Be aware of unconventional ways calculators can display an answereg????????8 > x > 2?rather than?2 < x < 8