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Quadratic Formula (Cambridge O Level Additional Maths)

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Mark

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Mark

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Quadratic Formula

How do I use the quadratic formula to solve a quadratic equation?

  • A quadratic equation has the form:

    ax2 + bx + c = 0 (as long as a ≠ 0)

    • you need "= 0" on one side
  • The quadratic formula is a formula that gives both solutions:
    • x equals fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction
  • Read off the values of a, b and c from the equation
  • Substitute these into the formula
    • write this line of working in the exam
    • Put brackets around any negative numbers being substituted in
  • To solve 2 x squared minus 5 x plus 2 equals 0 using the quadratic formula:
    • a = 2, b = -5 and c = 2
    • x equals fraction numerator negative open parentheses negative 5 close parentheses plus-or-minus square root of open parentheses negative 5 close parentheses squared minus 4 cross times 2 cross times 2 end root over denominator 2 cross times 2 end fraction
    • Carefully simplify by doing the calculation in parts
      x equals fraction numerator 5 plus-or-minus square root of 25 minus 16 end root over denominator 4 end fraction equals fraction numerator 5 plus-or-minus square root of 9 over denominator 4 end fraction
    • Separate the + and - to get the two answers
      table row x equals cell fraction numerator 5 plus 3 over denominator 4 end fraction equals 8 over 4 equals 2 end cell row x equals cell fraction numerator 5 minus 3 over denominator 4 end fraction equals 2 over 4 equals 1 half end cell end table
  • On the non-calculator paper, answers may be required in exact form
  • On the calculator paper, you might have to round your answers
    • Give your answers in exact form before rounding in case you round incorrectly

Examiner Tip

  • On the calculator paper you will be able to check your solutions using the quadratic equation solver feature if your calculator has one
  • Always look for how the question wants you to leave your final answers
    • for example, correct to 2 decimal places

Worked example

Use the quadratic formula, without a calculator, to find the exact solutions of the equation 12 x squared minus 17 x plus 6 equals 0.

Write down the values of a, b and c
 

a equals 12 comma space b equals negative 17 comma space c equals 6
 

Substitute these values into the quadratic formula, x equals fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction
Put brackets around any negative numbers
 

x equals fraction numerator negative open parentheses negative 17 close parentheses plus-or-minus square root of open parentheses negative 17 close parentheses squared minus 4 cross times 12 cross times 6 end root over denominator 2 cross times 12 end fraction
 

Simplify

table row x equals cell fraction numerator 17 plus-or-minus square root of 289 minus 288 end root over denominator 24 end fraction end cell row blank equals cell fraction numerator 17 plus-or-minus square root of 1 over denominator 24 end fraction end cell end table

Work out the + and - parts separately to get the two solutions 

x equals fraction numerator 17 plus 1 over denominator 24 end fraction equals 18 over 24 x equals fraction numerator 17 minus 1 over denominator 24 end fraction equals 16 over 24
 

bold italic x bold equals bold 3 over bold 4 bold comma bold space bold space bold italic x bold equals bold 2 over bold 3

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    Mark

    Author: Mark

    Expertise: Maths

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.