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 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.)
Transcript text: Solve the equation using the quadratic formula. \[ x^{2}+13 x+8=0 \] The solution set is $\square$ (Simplify your answer. Type an exact answer, using radicals and $i$ as needed. Use inte separate answers as needed.)
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Solution

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Solve the quadratic equation x2+13x+8=0x^{2} + 13x + 8 = 0 using the quadratic formula.

Identify the coefficients

The quadratic equation is in the form ax2+bx+c=0ax^{2} + bx + c = 0, where:

  • a=1a = 1
  • b=13b = 13
  • c=8c = 8

Apply the quadratic formula

Write the quadratic formula

The quadratic formula is: x=b±b24ac2a x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}

Substitute the coefficients into the formula

Substitute a=1a = 1, b=13b = 13, and c=8c = 8 into the formula: x=13±13241821 x = \frac{-13 \pm \sqrt{13^{2} - 4 \cdot 1 \cdot 8}}{2 \cdot 1}

Simplify under the square root

Calculate the discriminant: b24ac=132418=16932=137 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=13±1372 x = \frac{-13 \pm \sqrt{137}}{2}

Write the solution set

Express the solutions

The solutions to the equation are: x=13+1372andx=131372 x = \frac{-13 + \sqrt{137}}{2} \quad \text{and} \quad x = \frac{-13 - \sqrt{137}}{2}

The solution set is: {13+1372,131372} \boxed{\left\{ \frac{-13 + \sqrt{137}}{2}, \frac{-13 - \sqrt{137}}{2} \right\}}

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