Solving Quadratic Inequalities (Edexcel IGCSE Maths)

Spec point — 2.8A

Solving quadratic inequalities

What are quadratic inequalities?

• A quadratic inequality has the form $ax^{2}+bx+c\geq0$

  • There is an $x^{2}$ term and any inequality sign, $\geq,\leq,>,<$

  • They can usually be factorised

    • For example, $(x−2)(x−5) \ge 0$

• Solutions to quadratic inequalities are ranges of$x$values

  • For example, $1\leq x\leq4$

How do I solve a quadratic inequality?

• Quadratic inequalities are solved by sketching a graph

  • The solutions then appears along the $x$-axis

• For example, to solve $(x−2)(x−5) \ge 0$

  • Sketch the graph of y=(x−2)(x−5)

    • Show the $x$-intercepts of 2 and 5
    • These come from solving (x−2)(x−5)=0

  • As the inequality is $\geq0$, shade any parts of the curve that are above the$x$-axis

    • These are left of $x=2$ and right of $x=5$

  • The solutions are the ranges on the $x$-axis for those shaded parts

    • $x\leq2$ or $x\geq5$
    • Use the word "or" for two separate parts
    • In set notation this is $\{ x : x \le 2 \} \cup \{ x : x \ge 5 \}$

What do I do if the sign of the inequality changes?

• If the quadratic inequality is $ax^{2}+bx+c\geq0$ where $a$ is positive

  • then shade the parts of the curve above the $x$-axis

• If the quadratic inequality is $ax^{2}+bx+c\leq0$ where $a$ is positive

  • then shade the part of the curve below the $x$-axis

• For example, to solve $(x−2)(x−5) \le 0$

  • Repeat the process above but shade where the curve is below the $x$-axis

    • The solution is $2\leq x\leq5$
    • (You do not use the word "or", but could say $x\geq2$ "and" $x\leq5$)
    • In set notation this is $\{ x : x \ge 2 \} \cap \{ x : x \le 5 \}$, or alternatively $\{ x : 2 \le x \le 5 \}$

• If strict inequality signs are used, $>$ or $<$

  • then you must use strict inequalities in your final answer

How can quadratic inequalities be made harder?

• You may need to bring all the terms to one side first

  • For example $x^{2}-x<6$ becomes $x^{2}-x-6<0$

    • It is easier to pick the side with a positive $x^{2}$

• You may have to factorise the quadratic inequality first

  • This may involve factorising

    • $ax^{2}+bx+c$ into double brackets
    • a² x²−b²=(a x+b)(a x−b) (the difference of two squares)
    • a x²+b x=x (a x+b) (factorising out an $x$)

  • If it does not factorise, you may need the quadratic formula to find $x$-intercepts

• You may be given a quadratic inequality $ax^{2}+bx+c\geq0$ where $a$ is negative

  • This means the sketch will be an $\cap$ shape

• For example, solve $4-x^{2}>0$

  • The $x$ intercepts are found from $4-x^{2}=0$ giving $x=\pm2$
  • The same process is then applied, but the graph has an $\cap$ shape

Exam Hint Think of inequalities with $x^{2}$ terms as graph-sketching questions in the exam.

Worked Example Find the set of values for which $3x^{2}+2x-6>x^{2}+4x-2$

Answer:

Rearrange to get a zero on one side

$2x^{2}-2x-4>0$

Divide everything by 2 as it is a common factor

$x^{2}-x-2>0$

Find the roots of the quadratic

x squared minus x minus 2 equals 0 $> ( x - 2 ) ( x + 1 ) = 0$ x equals 2 comma space x equals negative 1

Sketch the graph and identify the parts which are above the x-axis

$x<-1\mathrm{or}x>2$