Completing the square

Completing the square is a method for solving for the roots of the general quadratic equation:

$ax^{2}+bx+c=0$, where $a\neq 0$

It is first taught using equations with "friendly" numbers in place of a, b, and c to get the student used to the process.

What one does is add and multiply by various carefully chosen constants to create an equation of the form:

$d^{2}x^{2}+2dex+e^{2}=f$ where d, e and f are constants expressed in a, b, and c.

This resolves to:

$(dx+e)^{2}=f$ (grouping)

$dx+e=\pm {\sqrt {f}}$ (take square root)

$dx=-e\pm {\sqrt {f}}$ (subtract e)

$x={\frac {-e\pm {\sqrt {f}}}{d}}$ (divide by e)

By then applying the process to the general equation, we can derive the quadratic formula:

$ax^{2}+bx+c=0$ (given)

$4a^{2}x^{2}+4abx+4ac=0$ (multiply by 4a)

$4a^{2}x^{2}+4abx=-4ac$ (subtract 4ac)

$4a^{2}x^{2}+4abx+b^{2}=-4ac+b^{2}$ (add b^2)

$(2ax+b)^{2}=b^{2}-4ac$ (group each side)

$2ax+b=\pm {\sqrt {b^{2}-4ac}}$ (take square root, allow for both roots)

$2ax=-b\pm {\sqrt {b^{2}-4ac}}$ (subtract b)

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

We can now determine the real or imaginary roots of any quadratic equation by simply inserting a, b, and c into the formula.