\( -5 x^{2}+6 x-2=0 \) Write one exact, simplified solution in each box. You can add or remove boxes. If a solution not a real number, write it in the form \( \mathrm{a}+\mathrm{bi} \) or \( \mathrm{a}-\mathrm{bi} \), where a and b are real numbers. +

To solve the quadratic equation: \[ -5x^{2} + 6x - 2 = 0 \] We can use the **quadratic formula**: \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] Where: - \( a = -5 \) - \( b = 6 \) - \( c = -2 \) **Step-by-Step Solution:** 1. **Calculate the discriminant (\( \Delta \)):** \[ \Delta = b^2 - 4ac = (6)^2 - 4(-5)(-2) = 36 - 40 = -4 \] 2. **Determine the nature of the roots:** Since the discriminant is negative (\( \Delta = -4 \)), the solutions will be complex numbers. 3. **Apply the quadratic formula:** \[ x = \frac{-6 \pm \sqrt{-4}}{2(-5)} = \frac{-6 \pm 2i}{-10} \] 4. **Simplify the expression:** \[ x = \frac{6 \mp 2i}{10} = \frac{3}{5} \mp \frac{1}{5}i \] **Final Solutions:** \[ \boxed{ \begin{aligned} &\frac{3}{5} + \frac{1}{5}i \\ &\frac{3}{5} - \frac{1}{5}i \end{aligned} } \] These are the two exact, simplified solutions to the quadratic equation.