Quadratic formula

The quadratic formula can be used when simpler methods of solving a quadratic equation do not work.

First, the quadratic equation must be reduced to this format:

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\!}

Then the coefficients a, b, and c can be substituted in the formula to find the solutions:

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}}

You can prove the formula the following way:

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\!}

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

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

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

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

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}}

This method of deriving the formula is done via completing the square.

You can assert that the formula is correct by substituting the formula in place of x in 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\!} 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.