x2=2x+2

To solve the equation \( x^2 = 2x + 2 \), we first rearrange it into standard form. We can do this by moving all terms to one side of the equation:

\[ x^2 - 2x - 2 = 0 \]

Next, we can use the quadratic formula to find the solutions. The quadratic formula states that for any equation of the form \( ax^2 + bx + c = 0 \), the roots can be found using:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In this case:

• \( a = 1 \)
• \( b = -2 \)
• \( c = -2 \)

Now we can substitute these values into the quadratic formula:

1. Calculate the discriminant (\( b^2 - 4ac \)):

\[ (-2)^2 - 4(1)(-2) = 4 + 8 = 12 \]

2. Substitute into the formula:

\[ x = \frac{-(-2) \pm \sqrt{12}}{2(1)} = \frac{2 \pm \sqrt{12}}{2} \]

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

\[ x = \frac{2 \pm 2\sqrt{3}}{2} = 1 \pm \sqrt{3} \]

Thus, the solutions to the equation \( x^2 = 2x + 2 \) are:

\[ x = 1 + \sqrt{3} \quad \text{and} \quad x = 1 - \sqrt{3} \]