Quadratic Inequalities (Cambridge O Level Additional Maths)

What are quadratic inequalities?

• They are similar to quadratic equations with the "=" replaced by one of <, >, ≤ or ≥
  • Just like equations such inequalities should be in a form such that 0 is on one side of the inequality
    • e.g. 
• Sketching a quadratic graph is essential to finding the correct solution(s)
  • Some modern calculators may be able to solve quadratic inequalities directly
    • You could use this to check your answer

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 the graph of the quadratic and label the roots
  As Step 1 makes the -squared term positive it will be -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 will be x < x₁ or x > x₂ 
  For ax² + bx + c < 0 you want the region below the x-axis - the solution will be x₁ < x < x₂

• Be careful:
  
  • avoid multiplying or dividing by a negative number

    if unavoidable, “flip” the inequality sign so < → >, ≥ → ≤, etc

  • do rearrange to make the x² term positive

   

Quadratic inequalities and the discriminant

• The discriminant of the quadratic function  is 
• It's value indicates the number of (real) roots the quadratic function has
  • if there are two roots
  • if  there is one root (repeated)
  • if  there are no roots
• The firsts and last of these are quadratic inequalities
• Some questions will require you to use the discriminant to set up and solve a quadratic inequality
  • For example: Find the values of such that the equation  has no real roots
    • Using the discriminant, and for no real roots,
    • Using the approach above, this leads to the quadratic inequality in , 
    • And using the method above, including sketching a graph, leads to the solutions  and 

What are inequalities on graphs?

• Inequalities can be represented on graphs by shaded regions and dotted or solid lines
• These inequalities have two variables, x and y
• Several inequalities are used at once
• The solution is an area on a graph (often called a region and labelled R)
• The inequalities can be linear or quadratic

How do I draw inequalities on a graph?

• Sketch each line or curve
  • If the inequality is strict (< or >) then use a dotted line
  • If the inequality is weak (≤ or ≥) then use a solid line
• Decide which side of the line satisfies the inequality
  • If unsure, choose a coordinate on one side and test it in the inequality
    • The origin is an easy point to use
  • If it satisfies the inequality then that whole side of the line satisfies the inequality
    • For example: (0,0) satisfies the inequality y < x² + 1 so you want the side of the curve that contains the origin