Use the quadratic formula to find exact solutions. \( 9 x^{2}+6 x=-5 \) \( x=\square \) (Simplify your answer. Type your answer in the form a + bi. Use a comma to separate answers as needed. Use integers or fractions for any numbers in the expression.)

To solve the quadratic equation \( 9x^2 + 6x + 5 = 0 \) using the quadratic formula, we first need to identify the coefficients \( a \), \( b \), and \( c \) from the standard form of a quadratic equation \( ax^2 + bx + c = 0 \). From the equation \( 9x^2 + 6x + 5 = 0 \): - \( a = 9 \) - \( b = 6 \) - \( c = 5 \) The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Now, we will calculate the discriminant \( b^2 - 4ac \): \[ b^2 - 4ac = 6^2 - 4 \cdot 9 \cdot 5 \] Calculating \( b^2 \): \[ 6^2 = 36 \] Calculating \( 4ac \): \[ 4 \cdot 9 \cdot 5 = 180 \] Now, substituting these values into the discriminant: \[ b^2 - 4ac = 36 - 180 = -144 \] Since the discriminant is negative, we will have complex solutions. Now we can substitute \( a \), \( b \), and the discriminant into the quadratic formula: \[ x = \frac{-6 \pm \sqrt{-144}}{2 \cdot 9} \] Calculating \( \sqrt{-144} \): \[ \sqrt{-144} = 12i \] Now substituting this back into the formula: \[ x = \frac{-6 \pm 12i}{18} \] We can simplify this expression: \[ x = \frac{-6}{18} \pm \frac{12i}{18} \] This simplifies to: \[ x = -\frac{1}{3} \pm \frac{2}{3}i \] Thus, the exact solutions are: \[ x = -\frac{1}{3} + \frac{2}{3}i, -\frac{1}{3} - \frac{2}{3}i \] So, the final answer is: \[ x = -\frac{1}{3} + \frac{2}{3}i, -\frac{1}{3} - \frac{2}{3}i \]