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 \( \} \) (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. There is no solution.
Ask by Chambers Pritchard. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solution set is \(\{-5, 3\}\).
Solution
To solve the quadratic equation \( z^{2} + 2z - 15 = 0 \) by factoring, follow these steps:
1. **Find two numbers that multiply to \(-15\) and add up to \(2\).**
- The numbers \(5\) and \(-3\) satisfy this condition because:
\[
5 \times (-3) = -15 \quad \text{and} \quad 5 + (-3) = 2
\]
2. **Factor the quadratic equation using these numbers:**
\[
z^{2} + 2z - 15 = (z + 5)(z - 3) = 0
\]
3. **Set each factor equal to zero and solve for \(z\):**
- \(z + 5 = 0 \Rightarrow z = -5\)
- \(z - 3 = 0 \Rightarrow z = 3\)
Therefore, the solution set is \(\{-5, 3\}\).
**Answer:**
**A. The solution set is \(\{-5, 3\}\)**
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Simplify this solution Mind Expander
To factor the equation \( z^{2}+2 z-15=0 \), we need to find two numbers that multiply to -15 and add to 2. The numbers 5 and -3 fit perfectly! We can express the equation as \( (z + 5)(z - 3) = 0 \). By applying the zero product property, we get \( z + 5 = 0 \) or \( z - 3 = 0 \). This gives us the solutions \( z = -5 \) and \( z = 3 \). Thus, the solution set is \( \{-5, 3\} \).
