Complete the square to solve the following quadratic equation: x2+4x−6=2 . Is the following correct: x=−2+23–√, −2−23–√ ? 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, let's follow the steps carefully.

1. First, rewrite the equation: \[ x^2 + 4x - 6 - 2 = 0 \] This simplifies to: \[ x^2 + 4x - 8 = 0 \]

2. Next, move the constant term to the other 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: \[ \left(\frac{4}{2}\right)^2 = 2^2 = 4 \] Add 4 to both sides: \[ x^2 + 4x + 4 = 8 + 4 \] This simplifies to: \[ (x + 2)^2 = 12 \]

4. Now, take the square root of both sides: \[ x + 2 = \pm \sqrt{12} \] We can simplify \( \sqrt{12} \) to \( 2\sqrt{3} \): \[ x + 2 = \pm 2\sqrt{3} \]

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

This gives us the solutions: \[ x = -2 + 2\sqrt{3}, \quad x = -2 - 2\sqrt{3} \]

Now, let's compare this with the solutions given in the question, \( x = -2 + \sqrt{23}, -2 - \sqrt{23} \).

The solutions we derived do not match the provided option. The given solution mentions \( \sqrt{23} \), while we used \( \sqrt{12} \) which simplifies to \( 2\sqrt{3} \).

Therefore, the answer is:

Enter 2 for No.