Question

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 ?
?

Enter 1 for Yes.

Enter 2 for No.

Answer 1

Let's solve the equation \( x^2 + 4x - 6 = 2 \) by completing the square.

First, we can rearrange the equation:

\[
x^2 + 4x - 6 - 2 = 0 \implies x^2 + 4x - 8 = 0
\]

Next, we complete the square for the expression \( x^2 + 4x \).

1. Take half of the coefficient of \( x \) (which is 4), square it, and add/subtract it inside the equation:

\[
\left(\frac{4}{2}\right)^2 = 2^2 = 4
\]

2. Rewrite the quadratic by adding and subtracting 4:

\[
x^2 + 4x + 4 - 4 - 8 = 0 \implies (x + 2)^2 - 12 = 0
\]

3. Move -12 to the other side:

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

4. Taking the square root of both sides gives us:

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

5. Simplifying \( \sqrt{12} = 2\sqrt{3} \):

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

6. Finally, solving for \( x \):

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

Thus, the solutions are \( x = -2 + 2\sqrt{3} \) and \( x = -2 - 2\sqrt{3} \).

The proposed solutions \( x = -2 + 2\sqrt{3} \) and \( x = -2 - 2\sqrt{3} \) are correct.

So, the answer is:

**1 for Yes.**