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Exercises Equations and inequalities

Example 4 from Algebra and Trigonometry, 2.5 Quadratic Equations

Solving a Quadratic Equation Using Grouping

Use grouping to factor and solve the quadratic equation: $4 x^{2} + 15 x + 9 = 0.$

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
First, multiply $a c : 4 (9) = 36.$ Then list the factors of $36.$

$\begin{array}{l} 1 \cdot 36 \\ 2 \cdot 18 \\ 3 \cdot 12 \\ 4 \cdot 9 \\ 6 \cdot 6 \end{array}$

The only pair of factors that sums to $15$ is $3 + 12.$ Rewrite the equation replacing the b term, $15 x,$ with two terms using 3 and 12 as coefficients of x. Factor the first two terms, and then factor the last two terms.

$\begin{matrix} 4 x^{2} + 3 x + 12 x + 9 & = & 0 \\ x (4 x + 3) + 3 (4 x + 3) & = & 0 \\ (4 x + 3) (x + 3) & = & 0 \end{matrix}$

Solve using the zero-product property.

$\begin{matrix} (4 x + 3) (x + 3) & = & 0 \\ (4 x + 3) & = & 0 \\ x & = & - \frac{3}{4} \\ (x + 3) & = & 0 \\ x & = & - 3 \end{matrix}$
The solutions are $- \frac{3}{4},$ and $−3.$ See Figure 3.

Figure 3

Your turn

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

Solve using factoring by grouping: $12 x^{2} + 11 x + 2 = 0.$

How did it go?

PreviousUsing the Zero-Product Property to Solve a Quadratic Equation Written as the Difference of Squares Next Solving a Polynomial of Higher Degree by Factoring

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 2.5, 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.