Quadratic Formula (Cambridge International IGCSE Further Maths)

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:

  • $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$

• 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 $2x^{2}-5x+2=0$ using the quadratic formula:

  • a = 2, b = -5 and c = 2
  • $x = \frac{- ( - 5 ) \pm \sqrt{( - 5 )^{2} - 4 \times 2 \times 2} }{2 \times 2}$
  • Carefully simplify by doing the calculation in parts $x=\frac{5\pm\sqrt{25-16}}{4}=\frac{5\pm\sqrt{9}}{4}$
  • Separate the + and - to get the two answers $\begin{gathered} x = \frac{5 + 3}{4} = \frac{8}{4} = 2 \\ x = \frac{5 - 3}{4} = \frac{2}{4} = 1 \frac{1}{2} \end{gathered}$

• 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

Exam Hint

• 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 $12x^{2}-17x+6=0$.

Write down the values of a, b and c * *

$a=12,b=-17,c=6$ * *

Substitute these values into the quadratic formula, $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ Put brackets around any negative numbers * *

$x = \frac{- ( - 17 ) \pm \sqrt{( - 17 )^{2} - 4 \times 12 \times 6} }{2 \times 12}$ * *

Simplify

$\begin{gathered} x = \frac{17 \pm \sqrt{289 - 288} }{24} \\ = \frac{17 \pm \sqrt{1} }{24} \end{gathered}$

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

$x=\frac{17+1}{24}=\frac{18}{24}\,x=\frac{17-1}{24}=\frac{16}{24}$ * *

$x=\frac{3}{4},x=\frac{2}{3}$