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 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 a \ne 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:

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

This resolves to:

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

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

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 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)

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 square 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.