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

To solve the equation \( x^2 + 4x - 6 = 2 \) by completing the square, we first rearrange the equation:

\[ x^2 + 4x - 6 - 2 = 0 \]

This simplifies to:

\[ x^2 + 4x - 8 = 0 \]

Next, we can move the constant term to the other side of the equation:

\[ x^2 + 4x = 8 \]

Now, we need to complete the square for the left-hand side. We take the coefficient of \( x \), which is \( 4 \), divide it by \( 2 \) to get \( 2 \), and then square it to get \( 4 \). We then add \( 4 \) to both sides:

\[ x^2 + 4x + 4 = 8 + 4 \]

This gives us:

\[ (x + 2)^2 = 12 \]

Next, we take the square root of both sides:

\[ x + 2 = \pm \sqrt{12} \]

Since \( \sqrt{12} = 2\sqrt{3} \), we have:

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

Now, we solve for \( x \):

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

Thus, the solutions are:

\[ x = -2 + 2\sqrt{3} \quad \text{and} \quad x = -2 - 2\sqrt{3} \]

The solutions you've written, \( x = -2 + 2\sqrt{3} \), \( -2 - 2\sqrt{3} \), are indeed correct.