Algebra - Quadratic Equations

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Section 2.5 : Quadratic Equations - Part I

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4. Solve the following quadratic equation by factoring.

\[19x = 7 - 6{x^2}\]

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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.

\[\begin{align*}6{x^2} + 19x - 7 & = 0\\ \left( {3x - 1} \right)\left( {2x + 7} \right) & = 0\end{align*}\] Show Step 2

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

\[\begin{array}{*{20}{c}}{3x - 1 = 0}\\{\displaystyle x = \frac{1}{3}}\end{array}\hspace{0.25in}{\mbox{OR}}\hspace{0.25in}\begin{array}{*{20}{c}}{2x + 7 = 0}\\{\displaystyle x = - \frac{7}{2}}\end{array}\]

Therefore the two solutions are : \(\require{bbox} \bbox[2pt,border:1px solid black]{{x = \frac{1}{3}\,\,{\mbox{and }}x = - \frac{7}{2}}}\)

We’ll leave it to you to verify that they really are solutions if you’d like to by plugging them back into the equation.