# The Quadratic Formula

Created | Updated Dec 22, 2008

The quadratic formula holds great history, but in 2003, the UK government decided to hold a debate to discuss whether to keep it in the UK’s GCSE national curriculum; thankfully it was saved. In this entry we will look at what it is and how to use it but please bear in mind that you will need a calculator.

### The Basics

Before we begin, the quadratic formula is:

*x=-b± √(b²-4ac)*

2a

2a

It is simply a method for working out quadratic equations and it gives two possible answers, hence the '*±*' symbol^{1}. A quadratic equation is used for many things in Physics like working out stopping distances but it can also be used in Biology too!

If this all looks too complicated, don't worry, all will soon be explained.

### The Creation

Now we have seen the quadratic formula, you probably want to know how it is created, well, here we go:

We start off with a quadratic equation — *ax²+bx+c=0* — where '*a*' is not equal to zero.

Divide through by '*a*'

*x² + x(b/a) + c/a = 0* or *x² + x(b/a) = -c/a*

Complete the square^{2}.

*(x + b/2a)² - (b/2a)² = -(c/a) or (x+b/2a)² = (b/2a)² -(c/a)*

Expand the brackets.

*(x + b/2a)² = b²/(4a²) - c/a or (x + b/2a)² = (b²-4ac)/(4a²)*

Take the square root of both sides.

*x + b/2a = ± √(b²-4ac/4a²) = ±√(b²-4ac)/2a*

Make '*x*' the subject of the equation.

*x = -b/2a ± √(b²-4ac)/2a*

Finally we get.

*x=-b± √(b²-4ac)*

2a

2a

An extra note to point out is:

If b² > 4ac then plans can proceed as normal.

If b² = 4ac = b² - 4ac = 0 then there is only one answer.

If b² < 4ac which will produce a negative number and your calculator won't like it as it will create a complex number.

### Example

A rectangle's length is 5 cm longer than its width; the area is 36 cm². Find the width of the rectangle.

Although a scientific calculator can work this out for you all in one go, you may one day be armed with only a normal one, so in which case the formula needs to be broken down into bite size chunks.

First things first, turn the question into the form *ax²+bx+c=0*. This may sound hard to do but using several steps will make it become much easier.

We already know the area of the rectangle and that the length is 5 cm bigger than the width. We also know that the area of a rectangle is its width times its length: *WL=A*. So if we substitute these values into a formula: *x(x+5)=36* This can then be re-arranged into the quadratic form *ax²+bx+c=0*:

*x(x+5) = 36*

Open the brackets.

*x² + 5x = 36*

Make the equation equal zero.

*x² + 5x +(-36) = 0*

Now we can use the quadratic formula to work out '*x*':

*x=-5 ± √(5²-4*1*-36)*

2*1

2*1

*x=-5 ± √(25²-(-144))*

2

2

*x=-5 ± 169*

2

2

*x=-5 ± 13*

2

2

We have now worked out '*x*' but, as warned, there are two answers. In most cases, like this one, one answer will be discarded because it is not possible. In this example, the two answers are 4 and -9 so the obvious answer would be 4 as it would be impossible for a rectangle to have a width of -9^{3}. In some instances, however, you must not discard one answer as it may be possible.

### Negatives

In cases where the quadratic equation has a negative, you must reverse the addition to subtraction, or vice versa, in the formula. For example, if the quadratic equation is *ax² - bx + c = 0* then the formula would look:

*x=b± √(b²-4ac)*

2a

2a

This shows that at the beginning, '*b*' is positive: it has reversed.

### Some Extra Help

If you want to learn a bit more about other methods for solving quadratic equations then The Table Method of Factorising Quadratic Expressions entry should help.

^{1}It means plus or minus.

^{2}Don't worry about this step if you don't understand it because it is a subject all on its own.

^{3}In the real world that is but in the theoretical one, well, that is a different story.