3. What values of \( x \) are solutions to: \[ 5 x^{2}-x-2=2 \] A -1 and \( \frac{4}{5} \) B 1 and \( \frac{-4}{5} \) C -1 and \( \frac{5}{4} \) D 1 and \( \frac{-5}{4} \)

First, we start by rearranging the equation \( 5 x^{2}-x-2=2 \) into standard form. By subtracting 2 from both sides, we get: \[ 5 x^{2} - x - 4 = 0. \] Now, we can apply the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 5 \), \( b = -1 \), and \( c = -4 \). Calculating the discriminant: \[ b^2 - 4ac = (-1)^2 - 4(5)(-4) = 1 + 80 = 81. \] Now substituting into the quadratic formula: \[ x = \frac{-(-1) \pm \sqrt{81}}{2(5)} = \frac{1 \pm 9}{10}. \] This gives us two potential solutions: 1. \( x = \frac{10}{10} = 1 \) 2. \( x = \frac{-8}{10} = -\frac{4}{5} \) Thus, the solutions are \( x = 1 \) and \( x = -\frac{4}{5} \), leading us to option B: 1 and \(-\frac{4}{5}\). Just like that, math problems can feel like little adventures, where the solution is just a few steps away! Happy calculating!