Quadratic Inequalities (Cambridge (CIE) IGCSE Additional Maths): Revision Note

Quadratic inequalities

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.  $a{x}^2+bx+c\le 0$

• 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 $x$-squared term positive it will be $\cup$-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 $a{x}^2+bx+c$ is $b^2-4ac$

• It's value indicates the number of (real) roots the quadratic function has

  • if $b^2-4ac>0$ there are two roots

  • if $b^2-4ac=0$ there is one root (repeated)

  • if $b^2-4ac<0$ 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 $k$ such that the equation $\left(k+1\right)x^2-4x+\left(k-2\right)=0$ has no real roots

    • Using the discriminant, and for no real roots, ${\left(-4\right)}^2-4\left(k+1\right)\left(k-2\right)<0$

    • Using the approach above, this leads to the quadratic inequality in $k$, $k^2-k-6>0$

    • And using the method above, including sketching a graph, leads to the solutions $k<-2$ and $k>3$

Inequalities on graphs

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