Question
Solve the following equation by completing the square. \[ x^{2}+2 x=15 \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is \( \left\{\begin{array}{l}\text { (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) } \\ \text { B. There is no solution. }\end{array}\right. \)
Ask by Stuart Rogers. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solution set is \( \{3, -5\} \).
Solution
To solve the equation \( x^{2} + 2x = 15 \) by completing the square, follow these steps:
1. **Move the constant term to the other side:**
\[
x^{2} + 2x = 15 \quad \Rightarrow \quad x^{2} + 2x - 15 = 0
\]
2. **Complete the square:**
- Take the coefficient of \( x \), which is 2, divide it by 2 to get 1, and then square it to get 1.
- Add and subtract this square inside the equation:
\[
x^{2} + 2x + 1 = 15 + 1 \quad \Rightarrow \quad (x + 1)^2 = 16
\]
3. **Take the square root of both sides:**
\[
x + 1 = \pm 4
\]
4. **Solve for \( x \):**
\[
x = -1 \pm 4
\]
- \( x = -1 + 4 = 3 \)
- \( x = -1 - 4 = -5 \)
**Solution Set:**
\[
\{3, -5\}
\]
**Answer:**
**A.** The solution set is \( \{3, -5\} \).
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Simplify this solution Bonus Knowledge
To solve the equation \( x^{2}+2x=15 \) by completing the square, we first move the constant term to the other side: \[ x^{2}+2x-15=0 \] Next, we complete the square for the expression on the left. Take half of the coefficient of \( x \) (which is 2), square it, and add it to both sides: 1. Half of 2 is 1, and \( 1^2 = 1 \). 2. Add 1 to both sides: \[ x^{2}+2x+1=15+1 \] Now we have: \[ (x+1)^2=16 \] Taking the square root of both sides gives us: \[ x+1=\pm 4 \] Now, solve for \( x \): 1. \( x+1=4 \) leads to \( x=3 \) 2. \( x+1=-4 \) leads to \( x=-5 \) Thus, the solution set is: A. The solution set is \( \left\{3, -5\right\} \)
