# The Quadratic Formula

Created | Updated Jan 9, 2009

Despite its history, the quadratic formula was nearly dropped from the UK's GCSE national curriculum – the UK government decided to hold a debate on whether to keep the formula, and thankfully it was saved. However, in case it does get dropped one day, this entry explains what the formula is and how to use it. Before reading any further, you may want to find a calculator and something to scribble on.

### The Basics

The quadratic formula is used to solve equations of the sort *ax ^{2} + bx + c = 0*

^{1}. The formula is:

*x=-b± √(b²-4ac)*

2a

2a

Note that the formula gives two possible solutions, hence the 'plus or minus' (*±*) symbol. A quadratic equation is used for many things in physics such as calculating stopping distances, but it can also be used in areas like biology too!

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

### Deriving the Formula

Having seen the quadratic formula, you probably want to know how it is derived. Well, here you go:

We start off with a quadratic equation (note that '*a*' is not equal to zero):

*ax*

^{2}+ bx + c = 0*x*(divide through by '

^{2}+ x(b/a) + c/a = 0*a*')

*x*(subtract '

^{2}+ x(b/a) = -c/a*c/a*')

*(x + b/2a)*(complete the square

^{2}- (b/2a)^{2}= -c/a^{2})

*(x+b/2a)*(add

^{2}= (b/2a)^{2}-c/a*(b/2a)*)

^{2}*(x + b/2a)*(expand the brackets)

^{2}= b^{2}/(4a^{2}) - c/a*(x + b/2a)*(move

^{2}= (b^{2}-4ac)/(4a^{2})*-c/a*inside the brackets)

*x + b/2a = ± √(b*(take the square root of both sides

^{2}-4ac/4a^{2})^{3})

*x + b/2a = ±√(b*(move

^{2}-4ac)/2a*4a*out of the brackets)

^{2}*x = -b/2a ± √(b*(make '

^{2}-4ac)/2a*x*' the subject of the equation)

Finally, we get:

*x=-b± √(b²-4ac)*

2a

2a

One thing worth pointing out is that the ± √(b^{2}-4ac) part decides the number of solutions available:

If b

^{2}- 4ac > 0 then there are two solutions to be found, and plans can proceed as normal.If b

^{2}- 4ac = 0 then there is only one solution, as the one will end up with-b/2a'plus or minus the square root of zero'.If b

^{2}- 4ac < 0 then there are no real solutions, as taking the square root of b^{2}- 4ac produces a complex number that will upset ordinary calculators and produce an error message.

### Using the Formula

A rectangle's length is 5 cm longer than its width and its area is 36 cm^{2}. Find the width of the rectangle.

Although a scientific calculator can work out this sort of thing for you with ease, you may well be armed with only a normal calculator or, in an exam, be asked to show your working.

First things first, turn the question into the form *ax ^{2}+bx+c=0*. This may sound tricky, but using several steps will make it much easier.

We know that the area of a rectangle is its width times its length: *(width).(length) = (area)*. Given that the length of the rectangle is 5 more than its width, we can substitute *'x'* for the width and *'x+5'* for the length. This produces the formula: *x(x+5)=36*, which can be rearranged into the quadratic form *ax ^{2}+bx+c=0*:

*x(x+5) = 36*

Open the brackets.

*x ^{2} + 5x = 36*

Make the equation equal zero.

*x ^{2} + 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 *b ^{2} - 4ac > 0*, 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

^{4}. In some instances, however, both answers may be possible.

### Negatives

In cases where the quadratic equation contains a negative, you must reverse the addition to subtraction, or vice versa, in the formula. For example, if the quadratic equation were *ax ^{2} - bx + c = 0* then the formula used would be:

*x=b± √(b²-4ac)*

2a

2a

The *- bx* in the equation represents a negative number, and this negative eliminates the minus sign in front of the quadratic formula's *-b*. It might also make sense to change *b ^{2}* to

*(-b)*, but the two are equivalent and so this is unnecessary.

^{2}### 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 should help.

^{1}These are often disguised as

*(px+q)(rx-s)*and so forth by crafty question-setters.

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

^{3}The ± sign is added as positive numbers have two roots – for instance, the roots of 100 are ten and minus ten, which can be written as ±10.

^{4}In the real world, that is – in the theoretical one, it would be a different story.