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      Solving quadratic equations by using the formula

      Lesson details

      Learning outcome

      I can solve quadratic equations algebraically by using the formula.

      Key learning points

      1. The quadratic formula can be derived from the general form of a quadratic equation.
      2. It can be derived by completing the square.
      3. The formula is useful when the quadratic cannot be easily factorised.
      4. Technology can be used to identify solutions and is useful for checking.

      Keywords

      • Quadratic formula - The quadratic formula is a formula for finding the solutions to any quadratic equation of the form ax^2 + bx + c = 0

      Common misconception

      Mistakes often occur when substituting into the quadratic formula with calculators, as pupils miss out brackets and change the priority of operations.

      Values should always be put in brackets when substituted. This avoids mistakes when squaring and subtracting negative values. It also makes it easier to change the values of a , b , and c when working on a new question.

      Teacher tip

      The first learning cycle derives the quadratic formula. The slides are animated to allow pupils to think about what they could do next before the next step appears on screen. For the task, you may wish to cut up the steps and allow students to move then round into the correct order.

      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

      6 Questions

      Q1.
      Using the formula $$3m = 20x + 12y$$, if $$x = 3$$ and $$y = 9$$, what calculation would find $$m$$?

      $$\frac{12 \times 3 + 20 \times 9}{3}$$
      Correct answer: $$\frac{20 \times 3 + 12 \times 9}{3}$$
      $$\frac{20 \times 3 + 12 \times 3}{9}$$

      Q2.
      Using the formula $$2m = 14x + y$$, if $$x = 8$$ and $$y = 5$$, what calculation would find $$m$$?

      Correct answer: $$\frac{14 \times 8 + 5}{2}$$
      $$\frac{5 \times 8 + 14}{2}$$
      $$\frac{14 \times 8 + 2}{5}$$

      Q3.
      Using the formula $$x = 3a + t^2$$, if $$a = 5$$ and $$t = 6$$, what calculation would find $$x$$?

      $$3 \times 5 + 2^6$$
      $$3 \times 6 + 5^2$$
      Correct answer: $$3 \times 5 + 6^2$$

      Q4.
      Using the formula $$x = 3a + t^2$$, if $$a = 3$$ and $$t = 1$$, what calculation would find $$x$$?

      Correct answer: $$3^2 + 1$$
      $$1^2 + 3$$
      $$3^2 + 11$$

      Q5.
      Rearrange $$3m + t + 12 = 14g$$ to make $$t$$ the subject

      Correct Answer: t = 14g - 3m - 12, t = 14g - 12 - 3m

      Q6.
      Rearrange $$3m - 5 = 14g - w$$ to make $$w$$ the subject

      Correct Answer: w = 14g - 3m + 5, w = 14g + 5 - 3m

      6 Questions

      Q1.
      What are the values of x that solve $$2x^2 + 12x + 10 = 0$$ ? (answer in the form "a,b")

      Correct Answer: -5, -1, -1, -5

      Q2.
      What are the values of x that solve $$x^2 - 6x + 8 = 0$$ ? (answer in the form "a,b")

      Correct Answer: 2,4, 4,2

      Q3.
      What are the values of x that solve $$3x^2 + 9x - 12 = 0$$ ? (answer in the form "a,b")

      Correct Answer: -4,1, 1,-4

      Q4.
      What are the values of x that solve $$x^2 + 4x - 21 = 0$$ ? (answer in the form "a,b")

      Correct Answer: -7,3, 3,-7

      Q5.
      What are the values of x that solve $$x^2 + 10x + 16 = 0$$ ? (answer in the form "a,b")

      Correct Answer: -8,-2, -2,-8

      Q6.
      What are the values of x that solve $$ x^2 + 6x + 8 = 0$$ ? (answer in the form "a,b")

      Correct Answer: -4,-2, -2,-4

      To help you plan your 10 maths lesson on: Solving quadratic equations by using the formula, download all teaching resources for free and adapt to suit your pupils' needs...