2x²-5x-3=0

To solve the quadratic equation \(2x^2 - 5x - 3 = 0\), you can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a = 2\), \(b = -5\), and \(c = -3\). 1. Calculate the discriminant (\(b^2 - 4ac\)): \[ b^2 - 4ac = (-5)^2 - 4(2)(-3) = 25 + 24 = 49 \] 2. Substitute the values into the quadratic formula: \[ x = \frac{-(-5) \pm \sqrt{49}}{2(2)} = \frac{5 \pm 7}{4} \] 3. This gives us two potential solutions: - For the positive case: \[ x = \frac{5 + 7}{4} = \frac{12}{4} = 3 \] - For the negative case: \[ x = \frac{5 - 7}{4} = \frac{-2}{4} = -\frac{1}{2} \] So, the solutions to the equation \(2x^2 - 5x - 3 = 0\) are: \[ x = 3 \quad \text{and} \quad x = -\frac{1}{2} \]