Questions: Solve the quadratic equation. -5x^2 + 6x - 2 = 0

Solution Steps

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 \(a = -5\), \(b = 6\), and \(c = -2\). We will calculate the discriminant (\(b^2 - 4ac\)) first, then find the two possible values for \(x\).

The given quadratic equation is: \[ -5x^2 + 6x - 2 = 0 \] Here, the coefficients are: \[ a = -5, \quad b = 6, \quad c = -2 \]

The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Substitute the values of \(a\), \(b\), and \(c\) into the formula: \[ x = \frac{-6 \pm \sqrt{6^2 - 4(-5)(-2)}}{2(-5)} \]

Calculate the discriminant: \[ b^2 - 4ac = 6^2 - 4(-5)(-2) = 36 - 40 = -4 \] Since the discriminant is negative, the solutions will be complex numbers.

\[ x = \frac{-6 \pm \sqrt{-4}}{-10} \] Simplify the square root of the negative number: \[ \sqrt{-4} = 2i \] Thus, the equation becomes: \[ x = \frac{-6 \pm 2i}{-10} \]

Separate the real and imaginary parts: \[ x = \frac{-6}{-10} \pm \frac{2i}{-10} = \frac{3}{5} \mp \frac{i}{5} \]

Final Answer