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Exercises Equations and inequalities

Example 10 from Algebra and Trigonometry, 2.6 Other Types of Equations

Solving an Equation in Quadratic Form Containing a Binomial

Solve the equation in quadratic form: ${(x + 2)}^{2} + 11 (x + 2) - 12 = 0.$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
This equation contains a binomial in place of the single variable. The tendency is to expand what is presented. However, recognizing that it fits the criteria for being in quadratic form makes all the difference in the solving process. First, make a substitution, letting $u = x + 2.$ Then rewrite the equation in u.

$\begin{matrix} u^{2} + 11 u - 12 & = & 0 \\ (u + 12) (u - 1) & = & 0 \end{matrix}$

Solve using the zero-factor property and then replace u with the original expression.

$\begin{matrix} u + 12 & = & 0 \\ u & = & −12 \\ x + 2 & = & −12 \\ x & = & −14 \end{matrix}$

The second factor results in

$\begin{matrix} u - 1 & = & 0 \\ u & = & 1 \\ x + 2 & = & 1 \\ x & = & −1 \end{matrix}$

We have two solutions: $−14,$ and $−1.$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Solve: ${(x - 5)}^{2} - 4 (x - 5) - 21 = 0.$

How did it go?

PreviousSolving a Fourth-degree Equation in Quadratic Form Next Solving a Rational Equation Leading to a Quadratic

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 2.6, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.