Where does the quadratic formula come from?

First of all, let us be sure we understand what is meant by the term

. To do that, we must understand what is meant by the termsquadratic formulaandquadratic expression.quadratic equationA

is a polynomial of this form:quadratic expressionA

is an equation of the form:quadratic equationExamples of quadratic equations would be:

0=6x^{2}+2x +9

0=3.2x^{2}+8.2x- 1.6When equations of this form are solved for

, you get you get thex, which is:quadratic formulaThat is, you can use this calculation to find the values of

that make the quadratic equation equal to zero. These specific values ofxare called thexof the quadratic equation. They are also called therootsof the equation.zerosLet us figure out why

is equal to the above value. First, start with the quadratic formula:x[1]

Subtract

from both sides to get:c[2]

Divide both sides by

:a[3]

This next step may look a bit mysterious at first. It will allow us to turn the left side of the above equation into a

. This method of turning one side of an equation into a perfect square is often calledperfect square.'completing the square'Add to both sides:

[4]

the left side of this equation by asking this question: What two numbers when added are and when multiplied are ?FactorThat number is .

Therefore, we can see the left side of [4] as a perfect square that factors this way:

[5]

The left side can now be written this way:

[6]

Let us do a bit of algebra on the right side of [6]:

[7]

Now, we get a common denominator on the right side:

[8]

Adding on the right side:

[9]

Taking the square root of each side:

[10]

Subtract from each side:

[11]

Do some algebra on the right side:

[12]

Since the terms on the right side now have a common denominator, we will combine those terms by adding them:

[13]

And we are all done. [13] is the solution to the quadratic equation.

The above document is thanks to Zona Land and can be found at http://id.mind.net/~zona/mmts/miscellaneousMath/quadraticFormula/quadraticFormula.html