Find the zeros of the function algebraically. f(x)=2x2−5x−25*f*(*x*)=2*x*2−5*x*−25 

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!

Find the zeros of the function algebraically. f(x) = 2x^2 - 5x - 25

Learn how to find the zeros of the quadratic function f(x) = 2x^2 - 5x - 25 using the quadratic formula. This step-by-step solution explains key algebraic concepts such as the discriminant and provides a clear breakdown of the quadratic equation. Ideal for students in Grades 8-10.

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