7-x2.+2x=0

To solve the equation \( 7 - x^2 + 2x = 0 \), let's rewrite it in a standard form:

\[ -x^2 + 2x + 7 = 0 \]

Now, let's rearrange it to make the x² term positive:

\[ x^2 - 2x - 7 = 0 \]

Next, we can use the quadratic formula to find the solutions. The quadratic formula is given by:

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

Here, \( a = 1 \), \( b = -2 \), and \( c = -7 \).

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

\[ b^2 - 4ac = (-2)^2 - 4(1)(-7) = 4 + 28 = 32 \]

2. Now substitute \( a \), \( b \), and the discriminant into the quadratic formula:

\[ x = \frac{-(-2) \pm \sqrt{32}}{2(1)} = \frac{2 \pm \sqrt{32}}{2} \]

3. Simplify \( \sqrt{32} \):

\[ \sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2} \]

Thus:

\[ x = \frac{2 \pm 4\sqrt{2}}{2} = 1 \pm 2\sqrt{2} \]

Now, we have two solutions:

\[ x = 1 + 2\sqrt{2} \quad \text{and} \quad x = 1 - 2\sqrt{2} \]

These are the solutions to the equation \( 7 - x^2 + 2x = 0 \).