Algebra - Quadratic Equations

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Section 2.5 : Quadratic Equations - Part I

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7. Solve the following quadratic equation by factoring.

\[12{x^2} = 25x\]

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The first thing we need to do is get everything on one side of the equation and then factor the quadratic.

\[\begin{align*}12{x^2} - 25x & = 0\\ x\left( {12x - 25} \right) & = 0\end{align*}\]

Make sure that you do not just cancel an \(x\) from both sides of the equation!

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Now all we need to do is use the zero factor property to get,

\[\begin{array}{*{20}{c}}{x = 0}\\{}\end{array}\hspace{0.25in}{\mbox{OR}}\hspace{0.25in}\begin{array}{*{20}{c}}{12x - 25 = 0}\\{\displaystyle x = \frac{{25}}{{12}}}\end{array}\]

Therefore the two solutions are : \(\require{bbox} \bbox[2pt,border:1px solid black]{{x = 0\,\,{\mbox{and }}x = \frac{{25}}{{12}}}}\)

Note that if we’d canceled an \(x\) from both sides of the equation in the first step we would have missed the solution \(x = 0\)!