The Quadratic Formula (Cambridge (CIE) IGCSE International Maths: Extended): Revision Note

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:

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

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

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

  • 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

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!

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

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

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

  • Substitute this back into the formula and simplify

    • $x=\frac{8\pm 2\sqrt{22}}{4}=\frac{2\left(4\pm \sqrt{22}\right)}{4}=\frac{4\pm \sqrt{22}}{2}$

    • 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 $\mathrm{i}$ 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!