5х^2-3х-5=0

To solve the quadratic equation \( 5x^2 - 3x - 5 = 0 \), you can use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

where \( a = 5 \), \( b = -3 \), and \( c = -5 \).

1. Calculate the discriminant (\( b^2 - 4ac \)): \[ b^2 - 4ac = (-3)^2 - 4(5)(-5) = 9 + 100 = 109 \]

2. Substitute the values into the quadratic formula: \[ x = \frac{-(-3) \pm \sqrt{109}}{2 \cdot 5} = \frac{3 \pm \sqrt{109}}{10} \]

Therefore, the solutions for the equation \( 5x^2 - 3x - 5 = 0 \) are:

\[ x = \frac{3 + \sqrt{109}}{10} \quad \text{and} \quad x = \frac{3 - \sqrt{109}}{10} \]