The Quadratic Formula (Cambridge International IGCSE Maths)

Quadratic Formula

What is the quadratic formula?

• A quadratic equation has the form ax² + bx + c = 0 (where a ≠ 0)

  • you need "= 0" on one side

• The quadratic formula is a formula that gives both solutions to a quadratic equation:

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

Exam Hint

• Make sure the quadratic equation has "= 0" on the right-hand side

  • Otherwise it needs rearranging first

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

• 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 2x² - 8x - 3 = 0 using the quadratic formula:

  • a = 2, b = -8 and c = -3

  • $x = \frac{- ( - 8 ) \pm \sqrt{( - 8 )^{2} - 4 \times 2 \times ( - 3 )} }{2 \times 2}$

  • Type this into a calculator or simplify by hand

    • Type it once using + for ± then again using - for ±

  • The solutions are x = 4.3452078... or x = -0.34520787....

    • To 3 decimal places: x = 4.345 or x = -0.345
    • To 3 significant figures: x = 4.35 or x = -0.345

Exam Hint

• Always look for how the question wants you to leave your final answers

  • For example, correct to 2 decimal places

How do I write the solutions in an exact (surd) form?

• You may be asked to give answers in an exact (surd) form

  • For example, in a non-calculator paper

• In the example above, work out the number under the square root sign

  • Be careful with negatives!

    • $( - 8 )^{2} - 4 \times 2 \times ( - 3 ) = 64 + 24 = 88$

  • Now square root this number and use surd rules to simplify

    • $\sqrt{88}=\sqrt{4\times22}=\sqrt{4}\times\sqrt{22}=2\sqrt{22}$

  • Substitute this back into the formula and simplify

    • $x = \frac{8 \pm 2 \sqrt{22} }{4} = \frac{2 ( 4 \pm \sqrt{22} )}{4} = \frac{4 \pm \sqrt{22} }{2}$
    • The solutions in exact (surd) form are $x=\frac{4+\sqrt{22}}{2}$ or $x=\frac{4-\sqrt{22}}{2}$

• Calculators that can solve quadratics will give solutions in exact (surd) form

What is the discriminant?

• The part of the formula under the square root (b² – 4ac) is called the discriminant

• The sign of this value tells you if there are 0, 1 or 2 solutions

  • If b²– 4ac > 0 (positive)

    • then there are 2 different solutions

  • If b²– 4ac = 0

    • then there is only 1 solution
    • sometimes called "two repeated solutions"

  • If b²– 4ac < 0 (negative)

    • then there are no solutions
    • If your calculator gives you solutions with $i$ terms in, these are "complex" and are not what we are looking for

  • Interestingly, if b² – 4ac is a perfect square number ( 1, 4, 9, 16, …) then the quadratic expression could have been factorised!

Worked Example Use the quadratic formula to find the solutions of the equation 3x² - 2x - 4 = 0. Give each solution as an exact value in its simplest form.

Write down the values of a, b and c

a = 3, b = -2, c = -4*

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

$x = \frac{- ( - 2 ) \pm \sqrt{( - 2 )^{2} - 4 \times 3 \times ( - 4 )} }{2 \times 3}$

Simplify the expressions

$x=\frac{2\pm\sqrt{4+48}}{6}=\frac{2\pm\sqrt{52}}{6}$

Simplify the surd

$x==\frac{2\pm\sqrt{4\times13}}{6}=\frac{2\pm2\sqrt{13}}{6}$

Simplify the fraction

$x=\frac{1\pm\sqrt{13}}{3}$