Question

Complete the square to solve the following quadratic equation: x2+4x−6=2
𝑥
2
+
4
𝑥
−
6
=
2
. Is the following correct: x=−2+23–√, −2−23–√
𝑥
=
−
2
+
2
3
,

−
2
−
2
3
?

Enter 1 for Yes.

Enter 2 for No.

(1 point)

Answer 1

To solve the quadratic equation \(x^2 + 4x - 6 = 2\) by completing the square, we can follow these steps:

1. First, rearrange the equation:
\[
x^2 + 4x - 6 - 2 = 0
\]
Simplifying this gives:
\[
x^2 + 4x - 8 = 0
\]

2. Next, move the constant to the right-hand side:
\[
x^2 + 4x = 8
\]

3. Now, complete the square on the left side. Take half of the coefficient of \(x\) (which is 4), square it, and add it to both sides:
\[
\left(\frac{4}{2}\right)^2 = 4
\]
So we add 4:
\[
x^2 + 4x + 4 = 8 + 4
\]
This simplifies to:
\[
(x + 2)^2 = 12
\]

4. Now, take the square root of both sides:
\[
x + 2 = \pm \sqrt{12}
\]

5. Simplify \(\sqrt{12}\):
\[
\sqrt{12} = 2\sqrt{3}
\]
So we have:
\[
x + 2 = \pm 2\sqrt{3}
\]

6. Finally, solve for \(x\):
\[
x = -2 \pm 2\sqrt{3}
\]

Therefore, the solutions are:
\[
x = -2 + 2\sqrt{3}, \quad x = -2 - 2\sqrt{3}
\]

The original proposed solutions were:
\[
x = -2 + 23\sqrt{}, \quad -2 - 23\sqrt{}
\]

This is not correct. The correct solutions involve \(2\sqrt{3}\), not \(23\).

The answer is:

**Enter 2 for No.**