Quadratic formula

The quadratic formula can be used when simpler methods of solving a quadratic equation do not work.

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.