## Derive Quadratic Formula

If you want an explanation on using the formula go to the quadratic formula post (with a downloadable solver)

The quadratic formula,

for the general equation ax^{2}+bx+c=0

can be derived using the method of “completing the square” as follows.

starting with

ax^{2}+bx+c=0

divide through by a

x^{2}+bx/a+c/a=0

take the c part to the other side

x^{2}+bx/a=-c/a

In order to complete the square we need to add half the coefficient of x (that’s b/2a) squared to both sides so we get

x^{2}+bx/a + b^{2}/4a^{2}=-c/a +b^{2}/4a^{2}

Now we can factorise the left into a squared bracket (you can check this by multiplying it back out)

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

So if we square root both sides and take b/2a to the other side we have x on its own

x = -b/2a ±√(-c/a +b^{2}/4a^{2})

Now to get this in the more traditional form we can take out 1/2a (Which means each term in the root is timesed by 4a^{2}) from the square root to put the whole thing over 2a

**x = (-b ±√(b ^{2}-4ac))/2a**

David Woodford

## Quadratic Formula

The quadratic formula is a quick(unless you can factorise) way of solving quadratic equations. You basically take the coefficient’s of x, x^{2} and numbers, put then in the formula, work out the two answers and have your 2 solutions of x. And if that wasn’t easy enough written a console program in c++ that will solve them for you(and gives dodgy answers for complex solution ie) imaginary answers when there are no real roots.

so for the general quadratic

ax^{2}+bx+c=0

So what do you do, well enter a, b and c from the general equation into the formula, work out the answers and they are your solutions.

Whats the plus/minus thingy. You might be wondering what the thing is after the -b, well its a plus or minus sign. Because when you work out a square root it can have to answers, eg)root 9 = +3 and -3, he formula takes this into account by saying you must use the plus and the minus answers. This therefore means you will be 2 solutions to the quadratic, which makes sense as the graph is a curve and it therefore must cut the x-axis twice.

When the root part is negative(before you find the root) there are no real roots, only complex ones. This means that the curve comes down above the x-axis and doesn’t cut it. If you work out the complex roots, using i for root(-1), your pair of answers will be a conjugate pair.

**David Woodfords Quadratic Calculator**

This is a console program that i wrote in c++ that will solve a quadratic for you. Just download the file and run it, follow the instructions and it will output the 2 answers for you. I tried to make it work for complex roots as well but somewhere in the decimal data types Ive messed up so only the second parts of the complex answers are correct – though im sure working out -b/2a isn’t too hard for you.

Download quadratic calculator

### Other posts relating to the quadratic formula

**Derive the Quadratic formula**: an explanation of how to derive and prove the quadratic formula by completing the square**Factorise Quadratics**: how to solve a quadratic without using the formula

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