To solve the quadratic equation \(2x^2 + 5x - 2 = 0\), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In this equation, \(a = 2\), \(b = 5\), and \(c = -2\).
First, we need to calculate the discriminant \(b^2 - 4ac\):
\[ b^2 = 5^2 = 25 \] \[ 4ac = 4 \times 2 \times -2 = -16 \] \[ b^2 - 4ac = 25 - (-16) = 25 + 16 = 41 \]
Now we can substitute the values into the quadratic formula:
\[ x = \frac{-5 \pm \sqrt{41}}{2 \times 2} = \frac{-5 \pm \sqrt{41}}{4} \]
Thus, the solutions for \(x\) are:
\[ x = \frac{-5 + \sqrt{41}}{4} \quad \text{and} \quad x = \frac{-5 - \sqrt{41}}{4} \]
These are the two solutions for the equation \(2x^2 + 5x - 2 = 0\).