Quadratic formula

The quadratic formula is the formula for finding the solutions of quadratic equations. Students are sometimes taught a method known as factoring, but that's really just looking for a lucky guess.

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:

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

The formula is derived in the following way, which is known as completing the square:

$ax^{2}+bx+c=0\!$

$x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}=0\!$

Now we need to get something of the form $(x+Q)^{2}\!$ that matches the first two terms. We have

$(x+Q)^{2}=x^{2}+2Qx+Q^{2}\!$

So we need $Q={\frac {b}{2a}}\!$ to get a match.

$(x+{\frac {b}{2a}})^{2}=x^{2}+{\frac {b}{a}}x+({\frac {b}{2a}})^{2}\!$

Plugging that in, we get

$(x+{\frac {b}{2a}})^{2}-({\frac {b}{2a}})^{2}+{\frac {c}{a}}=0\!$

$(x+{\frac {b}{2a}})^{2}=({\frac {b}{2a}})^{2}-{\frac {c}{a}}\!$

$(x+{\frac {b}{2a}})^{2}={\frac {b^{2}-4ac}{4a^{2}}}\!$

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

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

You can check that the formula is correct by substituting the formula (with either sign for the square root) 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.