Question
Solve the following equation by factoring. \[ z^{2}+2 z-15=0 \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is \( \{\square\} \). (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. There is no solution.
Ask by Wood Brewer. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solutions are \( z = -5 \) and \( z = 3 \).
Solution
To solve the quadratic equation by factoring:
\[
z^{2} + 2z - 15 = 0
\]
1. **Factor the quadratic:**
We look for two numbers that multiply to **-15** and add up to **2**. These numbers are **5** and **-3**.
\[
z^{2} + 2z - 15 = (z + 5)(z - 3) = 0
\]
2. **Find the solutions:**
Set each factor equal to zero:
\[
z + 5 = 0 \quad \Rightarrow \quad z = -5
\]
\[
z - 3 = 0 \quad \Rightarrow \quad z = 3
\]
**Answer:**
**A. The solution set is \( \{-5, 3\} \).**
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Simplify this solution Beyond the Answer
To solve the equation \( z^{2}+2z-15=0 \) by factoring, we need to find two numbers that multiply to \(-15\) (the constant term) and add up to \(2\) (the coefficient of \(z\)). The numbers \(5\) and \(-3\) satisfy these conditions. Thus, we can factor the equation as: \[ (z + 5)(z - 3) = 0 \] Setting each factor equal to zero gives us the solutions: \[ z + 5 = 0 \quad \Rightarrow \quad z = -5 \] \[ z - 3 = 0 \quad \Rightarrow \quad z = 3 \] Therefore, the solution set is \(\{-5, 3\}\). A. The solution set is \(\{-5, 3\}\\).
