Question

A) Complete the square to solve the following quadratic equation: x
2 + 4x = 41. (1 point)
x = − 2 + 5√3, − 2 − 5√3
x = 2 + 3√5, 2 − 3√5
x = − 2 + 3√5, − 2 − 3√5
x = 2 + 5√3, 2 − 5√3

Answer 1

To solve the equation \( x^2 + 4x = 41 \) by completing the square, we start by moving 41 to the left side:

\[
x^2 + 4x - 41 = 0
\]

Next, we focus on \( x^2 + 4x \). To complete the square, we take half of the coefficient of \( x \) (which is 4), square it, and add it to both sides of the equation. Half of 4 is 2, and squaring it gives us \( 2^2 = 4 \).

Now, we rewrite the equation:

\[
x^2 + 4x + 4 = 41 + 4
\]

This simplifies to:

\[
(x + 2)^2 = 45
\]

Next, we take the square root of both sides:

\[
x + 2 = \pm \sqrt{45}
\]

We simplify \( \sqrt{45} \):

\[
\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}
\]

So, we have:

\[
x + 2 = \pm 3\sqrt{5}
\]

Now, subtracting 2 from both sides gives us:

\[
x = -2 \pm 3\sqrt{5}
\]

Thus, the solutions are:

\[
x = -2 + 3\sqrt{5}, \quad x = -2 - 3\sqrt{5}
\]

Looking at the answer choices provided, the correct answer is:

**x = − 2 + 3√5, − 2 − 3√5**