Quadratic Inequalities (AQA GCSE Further Maths): Revision Note

Solving quadratic inequalities

What are quadratic inequalities?

• A quadratic inequality has the form $a{x}^2+bx+c\ge 0$

  • There is an $x^2$ term and any inequality sign, $\ge ,\le ,>,<$

  • They can usually be factorised

    • For example, $\left(x-2\right)\left(x-5\right)\ge 0$

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

  • For example, $1\le x\le 4$

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 $\left(x-2\right)\left(x-5\right)\ge 0$

  • Sketch the graph of $y=\left(x-2\right)\left(x-5\right)$

    • Show the $x$-intercepts of 2 and 5

    • These come from solving $\left(x-2\right)\left(x-5\right)=0$

  • As the inequality is $\ge 0$, 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

    • $x\le 2$ or $x\ge 5$

    • Use the word "or" for two separate parts

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

• If the quadratic inequality is $a{x}^2+bx+c\ge 0$ where $a$ is positive

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

• If the quadratic inequality is $a{x}^2+bx+c\le 0$ where $a$ is positive

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

• For example, to solve $\left(x-2\right)\left(x-5\right)\le 0$

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

    • The solution is $2\le x\le 5$

    • (You do not use the word "or", but could say $x\ge 2$ "and" $x\le 5$)

• 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

    • $a{x}^2+bx+c$ into double brackets

    • $a^2{x}^2-b^2=\left(ax+b\right)\left(ax-b\right)$ (the difference of two squares)

    • $a{x}^2+bx=x\left(ax+b\right)$ (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 $a{x}^2+bx+c\ge 0$ 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 $x=\pm 2$

  • The same process is then applied, but the graph has an $\cap$ shape