Use the quadratic formula to solve the following: \( 2 x^{2}-7 x-9=0 \)

To solve the equation \( 2x^2 - 7x - 9 = 0 \) using the quadratic formula, which is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), we first identify the coefficients: \( a = 2 \), \( b = -7 \), and \( c = -9 \). Now, plug these values into the formula: 1. Calculate the discriminant: \( b^2 - 4ac = (-7)^2 - 4(2)(-9) = 49 + 72 = 121 \). 2. Apply the quadratic formula: \( x = \frac{-(-7) \pm \sqrt{121}}{2(2)} \) \( x = \frac{7 \pm 11}{4} \). Calculating the two possible solutions: - For \( 7 + 11 \): \( x = \frac{18}{4} = \frac{9}{2} \) or 4.5. - For \( 7 - 11 \): \( x = \frac{-4}{4} = -1 \). Thus, the solutions are \( x = \frac{9}{2} \) (or 4.5) and \( x = -1 \).