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)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ax^2 + bx + c = 0} (given)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4a^2x^2 + 4abx + 4ac = 0} (multiply by 4a)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4a^2x^2 + 4abx = -4ac} (subtract 4ac)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4a^2x^2 + 4abx + b^2 = -4ac + b^2} (add b^2)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (2ax + b)^2 = b^2 - 4ac} (group each side)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2ax + b = \pm \sqrt{b^2 - 4ac}} (take sqaure root, allow for both roots)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2ax = -b \pm \sqrt{b^2 - 4ac}} (subtract b)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.