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      Solving quadratic inequalities algebraically

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      Lesson details

      Learning outcome

      I can solve a quadratic inequality algebraically.

      Key learning points

      1. The solutions to the quadratic equation can be found using one of your known methods
      2. By sketching the graph, it is easy to define the solution set
      3. By studying the features of the quadratic, it is easy to define the solution set

      Keywords

      • Inequality - An inequality is used to show that one expression may not be equal to another.

      • Quadratic - A quadratic is an equation, graph or sequence where the highest exponent of the variable is 2. The general form for a quadratic is ax^2 + bx + c

      Common misconception

      All inequalities are graphed with solid lines.

      When graphed, strict inequalities are indicated with a dashed line. This is important as it visually tells us that values on the line will not satisfy the inequality.

      Teacher tip

      Encourage pupils to identify the region that satisfies multiple inequalities in different ways: graphically, testing a point, algebraically.

      Licence

      This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0
       except where otherwise stated. See Oak's terms & conditions
       (Collection 2).

      Lesson video

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      Prior knowledge starter quiz

      Download starter quiz questions (PDF)

      6 Questions

      Q1.
      $$(x+8)(x-2)$$ is the quadratic $$x^2+6x-16$$ in __________ form.

      Correct answer: factorised
      solution
      expression
      equation
      inequality

      Q2.
      Match the quadratics to their factorised form.

      Correct Answer:$$x^2+15x+54$$,$$(x+9)(x+6)$$

      $$(x+9)(x+6)$$

      Correct Answer:$$x^2+3x-54$$,$$(x+9)(x-6)$$

      $$(x+9)(x-6)$$

      Correct Answer:$$x^2-3x-54$$,$$(x-9)(x+6)$$

      $$(x-9)(x+6)$$

      Correct Answer:$$x^2-15x+54$$,$$(x-9)(x-6)$$

      $$(x-9)(x-6)$$

      Correct Answer:$$2x^2+24x+54$$,$$(x+9)(2x+6)$$

      $$(x+9)(2x+6)$$

      Correct Answer:$$2x^2+21x+54$$,$$(2x+9)(x+6)$$

      $$(2x+9)(x+6)$$

      Q3.
      If $$x^2-7x-30=0$$ factorises to $$(x-10)(x+3)=0$$, where are its roots?

      Correct answer: $$x=10$$
      $$x=-10$$
      $$x=3$$
      Correct answer: $$x=-3$$
      $$x=-30$$

      Q4.
      Which of the below is the quadratic formula?

      Correct answer: $$x={{-b\pm{\sqrt{b^2-4ac}}}\over{2a}}$$
      $$x={{b\pm{\sqrt{b^2-4ac}}}\over{2a}}$$
      $$x=-b\pm{{\sqrt{b^2-4ac}}\over{2a}}$$
      $$x=-b+{{\sqrt{b^2-4ac}}\over{2a}}$$
      $$x={{-b+{\sqrt{b^2-4ac}}}\over{2a}}$$

      Q5.
      This is a sketch of $$y=x^2-10x+24$$. Use it to solve this inequality: $$x^2-10x+24<0$$.

      An image in a quiz
      $$x<6$$
      Correct answer: $$4<x<6$$
      $$10<x<24$$
      $$x<4$$ or $$x>6$$
      $$x>4$$ or $$x>6$$

      Q6.
      Which of these is a rearrangement of $$x^2-10x-87=0$$ when completing the square?

      $$(x-5)^2=87$$
      $$(x-5)^2=-87$$
      $$(x-5)^2=62$$
      $$(x-5)^2=-112$$
      Correct answer: $$(x-5)^2=112$$

      Assessment exit quiz

      Download exit quiz questions (PDF)

      6 Questions

      Q1.
      The curve $$y=x^2+2x-15$$ has __________ at $$x=-5$$ and $$x=3$$.

      solutions
      equations
      Correct answer: roots
      turning points

      Q2.
      Solve $$x^2+2x-15<0$$.

      $$x<3$$
      $$x<-5$$
      Correct answer: $$-5<x<3$$
      $$x<-5$$ and $$x>3$$
      There is no solution to this inequality.

      Q3.
      Solve $$x^2+2x-15>0$$.

      $$x>0$$
      Correct answer: $$x<-5$$ or $$x>3$$
      $$-5<x<3$$
      $$x<-5$$ or $$x<3$$
      There is no solution to this inequality.

      Q4.
      Solve $$x^2-9<16$$.

      Correct answer: $$-5<x<5$$
      $$x<-5$$ and $$x>5$$
      $$-3<x<3$$
      $$x<-3$$ and $$x>3$$
      There is no solution to this inequality.

      Q5.
      Solve $$-x^2-2x+15>0$$.

      $$x<-5$$ and $$x>3$$
      $$x<-3$$ and $$x>5$$
      $$x>-5$$ and $$x>3$$
      Correct answer: $$-5<x<3$$
      $$-3<x<5$$

      Q6.
      The solution to $$2x^2-5x+20<x^2+8x-10$$ is $$3<x<$$ .

      Correct Answer: 10

      To help you plan your 11 maths lesson on: Solving quadratic inequalities algebraically, download all teaching resources for free and adapt to suit your pupils' needs...