Question
Solve the equation given by completing the square. \( 5 x^{2}+30 x+25=0 \) [Hint: Divide by 5 first] \( x=\square \)
Ask by Christensen Floyd. in the United States
Jan 31,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x = -1 \) or \( x = -5 \)
Solution
To solve the equation \( 5x^{2} + 30x + 25 = 0 \) by completing the square, follow these steps:
1. **Divide the entire equation by 5** to simplify:
\[
x^{2} + 6x + 5 = 0
\]
2. **Move the constant term to the other side**:
\[
x^{2} + 6x = -5
\]
3. **Complete the square**:
- Take half of the coefficient of \( x \), which is \( 6/2 = 3 \).
- Square it: \( 3^2 = 9 \).
- Add 9 to both sides of the equation:
\[
x^{2} + 6x + 9 = -5 + 9
\]
\[
(x + 3)^2 = 4
\]
4. **Take the square root of both sides**:
\[
x + 3 = \pm 2
\]
5. **Solve for \( x \)**:
- When using the positive square root:
\[
x + 3 = 2 \implies x = -1
\]
- When using the negative square root:
\[
x + 3 = -2 \implies x = -5
\]
**Final Solutions:**
\[
x = -1 \quad \text{and} \quad x = -5
\]
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution Extra Insights
First, let's divide the entire equation by 5 to simplify it: \[ x^2 + 6x + 5 = 0 \] Next, we can move the constant term to the right side: \[ x^2 + 6x = -5 \] Now, we want to complete the square. To do this, take half of the coefficient of \(x\) (which is 6), square it (which gives us 9), and add it to both sides: \[ x^2 + 6x + 9 = 4 \] Now, we can express the left side as a perfect square: \[ (x + 3)^2 = 4 \] Taking the square root of both sides, we get: \[ x + 3 = \pm 2 \] Now, let's solve for \(x\): 1. \(x + 3 = 2\) leads to \(x = -1\) 2. \(x + 3 = -2\) leads to \(x = -5\) So, the solutions are: \[ x = -1 \quad \text{and} \quad x = -5 \]
