Find the roots of the polynomial equation using the quadratic formula: \( 2x^{2} + 3x - 2 = 0 \)

To find the roots of the polynomial equation \( 2x^{2} + 3x - 2 = 0 \), you can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 2 \), \( b = 3 \), and \( c = -2 \). Plugging in these values, you first calculate the discriminant: \[ b^2 - 4ac = 3^2 - 4 \times 2 \times (-2) = 9 + 16 = 25 \] Now, substitute back into the formula: \[ x = \frac{-3 \pm \sqrt{25}}{2 \times 2} = \frac{-3 \pm 5}{4} \] This gives us two potential solutions: 1. \( x = \frac{-3 + 5}{4} = \frac{2}{4} = \frac{1}{2} \) 2. \( x = \frac{-3 - 5}{4} = \frac{-8}{4} = -2 \) Thus, the roots of the equation \( 2x^{2} + 3x - 2 = 0 \) are \( x = \frac{1}{2} \) and \( x = -2 \). Now you've solved the equation, and you have two special numbers you can throw into your next math conversation to impress your friends!