Question

Complete the square to solve the following quadratic equation: x2+4x=41
.(1 point)
Responses

x=−2+35–√, −2−35–√
x equals negative 2 plus 3 Start Root 5 End Root comma negative 2 minus 3 Start Root 5 End Root

x=2+53–√, 2−53–√
x equals 2 plus 5 Start Root 3 End Root comma 2 minus 5 Start Root 3 End Root

x=2+35–√, 2−35–√
x equals 2 plus 3 Start Root 5 End Root comma 2 minus 3 Start Root 5 End Root

x=−2+53–√, −2−53–√

Answer 1

To solve the quadratic equation \( x^2 + 4x = 41 \) by completing the square, follow these steps:

1. **Move the constant to the right side**:
\[
x^2 + 4x - 41 = 0
\]
or equivalently,
\[
x^2 + 4x = 41
\]

2. **Complete the square**:
To complete the square for the left side, take half of the coefficient of \( x \) (which is 4), square it, and add it to both sides. Half of 4 is 2, and squaring it gives \( 2^2 = 4 \).
\[
x^2 + 4x + 4 = 41 + 4
\]
\[
(x + 2)^2 = 45
\]

3. **Solve for \( x \)**:
Take the square root of both sides:
\[
x + 2 = \pm \sqrt{45}
\]
Simplify \( \sqrt{45} \):
\[
\sqrt{45} = \sqrt{9 \cdot 5} = 3\sqrt{5}
\]
Now rewrite the equation:
\[
x + 2 = \pm 3\sqrt{5}
\]
Subtract 2 from both sides:
\[
x = -2 \pm 3\sqrt{5}
\]

4. **Final solutions**:
Thus, the solutions are:
\[
x = -2 + 3\sqrt{5} \quad \text{and} \quad x = -2 - 3\sqrt{5}
\]

The correct response is:
\[
x = -2 + 3\sqrt{5}, -2 - 3\sqrt{5}
\]