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Quadratic Formula (Cambridge (CIE) O Level Additional Maths): Revision Note

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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 Tips and Tricks

  • 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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