Solve the given equation step by step.

Let's solve the equation \( \frac{x^2 - 5}{2} - \frac{x - 8}{5} = 3 \). 1. Eliminate the fractions by finding the least common multiple (LCM) of the denominators (2 and 5), which is 10. Multiply through by 10: \[ 10 \cdot \left( \frac{x^2 - 5}{2} \right) - 10 \cdot \left( \frac{x - 8}{5} \right) = 10 \cdot 3. \] 2. Simplify each term: \[ 5(x^2 - 5) - 2(x - 8) = 30. \] 3. Expand the terms: \[ 5x^2 - 25 - 2x + 16 = 30. \] 4. Combine like terms: \[ 5x^2 - 2x - 9 = 30. \] 5. Subtract 30 from both sides to set the equation to 0: \[ 5x^2 - 2x - 39 = 0. \] 6. This is a quadratic equation. Solve it using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \] where \( a = 5 \), \( b = -2 \), \( c = -39 \). 7. Calculate the discriminant \( \Delta = b^2 - 4ac \): \[ \Delta = (-2)^2 - 4(5)(-39) = 4 + 780 = 784. \] 8. Calculate the roots: \[ x = \frac{-(-2) \pm \sqrt{784}}{2 \cdot 5} = \frac{2 \pm 28}{10}. \] 9. Simplify the roots: \[ x_1 = \frac{2 + 28}{10} = 3, \] \[ x_2 = \frac{2 - 28}{10} = -2.6. \] Thus, the solutions are \( x_1 = 3 \) and \( x_2 = -2.6 \).