Complete the square to solve the following quadratic equation: x2+4x−6=2

Question

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.