Questions: Solve the equation using the quadratic formula. x^2+13x+8=0 The solution set is (Simplify your answer. Type an exact answer, using radicals and i as needed. Use integers or fractions for any numbers in the expression. Separate answers as needed.)

Solve the quadratic equation \(x^{2} + 13x + 8 = 0\) using the quadratic formula.

Identify the coefficients

The quadratic equation is in the form \(ax^{2} + bx + c = 0\), where:

• \(a = 1\)
• \(b = 13\)
• \(c = 8\)

Apply the quadratic formula

Write the quadratic formula

The quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \]

Substitute the coefficients into the formula

Substitute \(a = 1\), \(b = 13\), and \(c = 8\) into the formula: \[ x = \frac{-13 \pm \sqrt{13^{2} - 4 \cdot 1 \cdot 8}}{2 \cdot 1} \]

Simplify under the square root

Calculate the discriminant: \[ b^{2} - 4ac = 13^{2} - 4 \cdot 1 \cdot 8 = 169 - 32 = 137 \]

Write the final expression

Substitute the discriminant back into the formula: \[ x = \frac{-13 \pm \sqrt{137}}{2} \]

Write the solution set

Express the solutions

The solutions to the equation are: \[ x = \frac{-13 + \sqrt{137}}{2} \quad \text{and} \quad x = \frac{-13 - \sqrt{137}}{2} \]

The solution set is: \[ \boxed{\left\{ \frac{-13 + \sqrt{137}}{2}, \frac{-13 - \sqrt{137}}{2} \right\}} \]