Find the roots of the function \(f(x) = 2x^2 - 5x - 3\).

Step 1 ：To find the roots of the function, we need to solve for \(x\) in the equation \(f(x) = 2x^2 - 5x - 3 = 0\).

Step 2 ：We will use the quadratic formula, which is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). In this case, \(a = 2\), \(b = -5\), and \(c = -3\).

Step 3 ：Substituting these values into the quadratic formula gives \(x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4*2*(-3)}}{2*2}\).

Step 4 ：Simplifying the above expression gives \(x = \frac{5 \pm \sqrt{25 + 24}}{4}\).

Step 5 ：Further simplifying gives \(x = \frac{5 \pm \sqrt{49}}{4}\).

Step 6 ：Taking the square root of 49 gives 7, so \(x = \frac{5 \pm 7}{4}\).

Step 7 ：This gives two possible solutions for \(x\), which are \(x = \frac{5 + 7}{4} = 3\) and \(x = \frac{5 - 7}{4} = -0.5\).