Quadratic formula

The quadratic formula is a general expression for the solutions to a quadratic equation. It is used when other methods, such as completing the square, factoring, and square root property do not work or are too tedious.

General Solution For A Quadratic by Completing the Square

We start with

\[ax^{2}+bx+c=0\]

Divide by $a$:

\[x^{2}+\frac{b}{a}x+\frac{c}{a}=0\]

Add $\frac{b^{2}}{4a^{2}}$ to both sides in order to complete the square:

\[\left(x^{2}+\frac{b}{a}x+\frac{b^{2}}{4a^{2}}\right)+\frac{c}{a}=\frac{b^{2}}{4a^{2}}\]

Complete the square:

\[\left(x+\frac{b}{2a}\right)^{2}+\frac{c}{a}=\frac{b^{2}}{4a^{2}}\]

Move $\frac{c}{a}$ to the other side:

\[\left(x+\frac{b}{2a}\right)^{2}=\frac{b^{2}}{4a^{2}}-\frac{c}{a}=\frac{ab^{2}-4a^{2}c}{4a^{3}}=\frac{b^{2}-4ac}{4a^{2}}\]

Take the square root of both sides:

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

Finally, move the $\frac{b}{2a}$ to the other side:

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

This is the quadratic formula, and we are done.

Video Solution by ligonmathkid2

https://youtu.be/Akz8LcVGj5k

Variation

In some situations, it is preferable to use this variation of the quadratic formula:

\[\frac{2c}{-b\pm\sqrt{b^2-4ac}}\]

See Also

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