9.5.4 Finding Exact Solutions

Algebra 19.5.4 Finding Exact Solutions

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For the equation $x^{2} + 14 x = - 46$ , the approximate solutions are $x \approx - 8.732$ and $x \approx - 5.268$ .

Find the exact solutions by completing the square. Show your reasoning.

Compare your answer:

$\begin{array}{rcl} x^{2} + 14 x & = & - 46 \\ x^{2} + 14 x + 49 & = & - 46 + 49 \\ x^{2} + 14 x + 49 & = & 3 \\ (x + 7)^{2} & = & 3 \\ \sqrt{(x + 7)^{2}} & = & \sqrt{3} \\ x + 7 & = & \pm \sqrt{3} \\ x & = & - 7 \pm \sqrt{3} \end{array}$

Why Should I Care?

A hand holds a tablet displaying a diagram of a revenue projection graph. The graph analyzes the revenue generated by different ticket prices.

The Ensemble Theatre is exploiting two ancient practices in new and exciting ways. Those ancient practices are quadratic equations and live theatre. Running a live theatre company requires many things, and one of the most important is the old tool called the quadratic function.

The theatre can use quadratic functions to determine ticket price to maximize the profit. Ticket prices need to be set high enough to recoup the cost of expenses and generate revenue while still being a price point that attracts customers.

The vertex of the parabola that represents the Ensemble Theatre's ticket price can tell us at what ticket price they can sell the most tickets and maximize the revenue.

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- Book title: Algebra 1
- Publication date: Jun 4, 2025
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