Question

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.

Answer 1

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

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

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

Next, we complete the square for the expression \( x^2 + 4x \). To do this, take half of the coefficient of \( x \) (which is 4), square it, and add and subtract that value inside the equation.

1. Half of 4 is 2. Squaring it gives \( 2^2 = 4 \).
2. We rewrite the equation:

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

This simplifies to:

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

Now, we isolate the squared term:

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

Taking the square root of both sides gives:

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

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

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

Now, solve for \( x \):

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

Thus, the solutions are:

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

Now, comparing with the given solutions \( x = -2 + \sqrt{23} \) and \( x = -2 - \sqrt{23} \), we can see that:

The correct solutions are:

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

Therefore, the statement that \( x = -2 + \sqrt{23}, -2 - \sqrt{23} \) is incorrect.

The answer is 2 (No).