The Quadratic Formula
Created | Updated Sep 10, 2013
For a look at the history of the quadratic equation check out The History Behind The Quadratic Formula
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The quadratic equation is probably one of the better known mathmatical formulas. It has been topic of many debates in newspapers, described as cruel torture by some, and as a fundamental part of human society in others. But thanks the a UK parliament debate in 2003 the formula is still part of the UK secondary school's curriculum
The quadratic formula is simply a method of solving those quadratic equations. To use it you first need a quadratic equation, this is a formula that has an x2 value in it, you then rearrange the formula into the form ax2+bx+c=0, such as 3x2+2x+35. After that it's just a matter of using some 'simple' algebra to substitue the a, b, c and x values in the quadratic formula, which is:
x=-b √(b2 -4ac)
2a
Please note: It is recommended you use a calculator to do this so as to avoid serious headaches.
All the bits you don't understand about what's said above
As with most of maths, the quadratic formula is easy to understand if you know what all the bits mean.
The first part you may not understand is the ± sign; this is the plus-minus sign. As you may know, quadratic formulas will almost always have two answers, for example, 36 is equal to both 62 and -62. The use of this formula means that you will solve the equation twice, one of the times round you will + at that point and the other time you will -.
The rest of it is easy to understand if you follow general algebraic rules.
An example
Consider this, a rectangles length is 5cm longer than its width. The area of the rectangle is 36cm2 and you need to find the width of the rectangle you are unable to work it out in your head and so, armed with paper, pen and calculator, you use the quadratic formula.
You first need to form an equation and put it into the form x2+bx+c=0. This may sound hard to do at first but using several steps it becomes much easier. You will know that the area of a rectangle is its width times its length: wl=A. You know the area and the width, and you also know that the length is the width plus 5. You substitute these values into the formula and get: x(x+5)=36. This is then re-arranged into the quadratic form ax2+bx+c=0:
x(x+5)=36
x2+5x=36
x25x-36=0
Now you can use the quadratic formula to find the values of x:
x=-b ±√(b2 -4ac)
2a
x=-5 ±√(52 -4*1*-36)
2*1
x=-5 ±√(25--144)
2
x=-5 ±√169
2
x=-5 ±13
2
x=4
x=-9
You now know that x can be equal to either 4 or -9. But it is not possible for the rectangle to have widths of two different values at the same time so you will need to discard one of the answers. In most cases, like this one, you simply look at the answers and notice that one of them cannot possibly be correct. In this question it is the -9 because it is not possible for a rectangle to have a width of -9. This leaves you with the answer that will work. But remember that this does not make the discarded answer to be incorrect in the theoretical world, just in the real one.
