Quadratic formula

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The quadratic formula is used to simplify the process of solving a quadratic equation.

First, the quadratic equation must be reduced to this format:

ax^2+bx+c=0\!

Then the coefficients a, b, and c can be substituted in the formula to find the solutions:

x = \frac{-b \pm \sqrt {b^2-4ac}}{2a}

You can prove the formula the following way:

ax^2+bx+c=0\!
x^2+\frac{b}{a}x+\frac{c}{a}=0\!
(x+\frac{b}{2a})^2-(\frac{b}{2a})^2+\frac{c}{a}=0\!
(x+\frac{b}{2a})^2=(\frac{b}{2a})^2-\frac{c}{a}\!
(x+\frac{b}{2a})^2=\frac{b^2-4ac}{4a^2}\!
x+\frac{b}{2a}=\frac{\pm \sqrt {b^2-4ac}}{2a}\!
x = \frac{-b \pm \sqrt {b^2-4ac}}{2a}

This method of deriving the formula is done via completing the square.

You can assert that the formula is correct by substituting the formula in place of x in ax^2+bx+c=0\! and then gradually simplifying the rather complicated formula that results, step by step. Eventually, if all the steps are done correctly, it will simplify to 0.

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