7x^2 + 8x + 7 = 0

To solve the quadratic equation \( 7x^2 + 8x + 7 = 0 \), we can use the quadratic formula:

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

where \( a = 7 \), \( b = 8 \), and \( c = 7 \).

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

\[ b^2 = 8^2 = 64 \] \[ 4ac = 4 \times 7 \times 7 = 196 \] \[ b^2 - 4ac = 64 - 196 = -132 \]

2. Since the discriminant is negative \((-132)\), the equation has no real solutions, but two complex solutions.

3. Now, substitute back into the quadratic formula:

\[ x = \frac{-8 \pm \sqrt{-132}}{2 \times 7} \] \[ = \frac{-8 \pm \sqrt{132}i}{14} \]

4. Simplify \(\sqrt{132}\):

\[ \sqrt{132} = \sqrt{4 \times 33} = 2\sqrt{33} \]

5. So the solutions are:

\[ x = \frac{-8 \pm 2\sqrt{33}i}{14} \] \[ = \frac{-4 \pm \sqrt{33}i}{7} \]

Thus, the two complex solutions are:

\[ x = \frac{-4 + \sqrt{33}i}{7} \quad \text{and} \quad x = \frac{-4 - \sqrt{33}i}{7} \]