We are tasked with finding the zeros of the quadratic function:

$f(x) = 2x^2 - 5x - 25$

Steps:

1. Set the function equal to zero: $2x^2 - 5x - 25 = 0$ We will solve this quadratic equation using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here, $a = 2$, $b = -5$, and $c = -25$.

2. Calculate the discriminant $\Delta$ (the expression under the square root): $\Delta = b^2 - 4ac = (-5)^2 - 4(2)(-25) = 25 + 200 = 225$

3. Substitute the values into the quadratic formula: $x = \frac{-(-5) \pm \sqrt{225}}{2(2)}$ $x = \frac{5 \pm 15}{4}$

4. Solve for the two possible values of $x$:

   • First solution: $x_1 = \frac{5 + 15}{4} = \frac{20}{4} = 5$
   • Second solution: $x_2 = \frac{5 - 15}{4} = \frac{-10}{4} = -\frac{5}{2}$

The zeros of the function are:

$x = 5 \quad \text{and} \quad x = -\frac{5}{2}$

Would you like further details or explanations?

Here are five related questions to explore:

1. What is the significance of the discriminant in solving quadratic equations?
2. Can all quadratic equations be solved using the quadratic formula?
3. How do you factor a quadratic equation without using the quadratic formula?
4. What does it mean graphically when a quadratic equation has two distinct solutions?
5. How do you find the vertex of a quadratic function?

Tip: Always check the discriminant before solving a quadratic equation. If it's negative, the quadratic will have no real solutions, only complex ones!