\[ 5 x^{2}+3 x=3 \] Step 2 of 2 : Use the discriminant, \( b^{2}-4 a c \), to determine the number of solutions of the given quadratic equation. Then formula \( x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} \)

To determine the number of solutions for the equation \( 5x^2 + 3x - 3 = 0 \), we first need to identify the coefficients \( a = 5 \), \( b = 3 \), and \( c = -3 \). Plugging these values into the discriminant formula, we get \( b^2 - 4ac = 3^2 - 4 \times 5 \times (-3) = 9 + 60 = 69 \). Since the discriminant is positive, this means there are two distinct real solutions! Now, let's solve for \( x \) using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-3 \pm \sqrt{69}}{10} \] This gives us two solutions: \( x_1 = \frac{-3 + \sqrt{69}}{10} \) and \( x_2 = \frac{-3 - \sqrt{69}}{10} \).