To find the zeros of the quadratic function $f(x) = 2x^2 + 18x + 33.8$, we can use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For the given function, we have:

• $a = 2$
• $b = 18$
• $c = 33.8$

Step 1: Calculate the discriminant

The discriminant $\Delta$ is given by:

$\Delta = b^2 - 4ac = (18)^2 - 4(2)(33.8)$

$\Delta = 324 - 4(2)(33.8) = 324 - 270.4 = 53.6$

Step 2: Apply the quadratic formula

Now substitute the values into the quadratic formula:

$x = \frac{-18 \pm \sqrt{53.6}}{2(2)}$

First, calculate $\sqrt{53.6}$:

$\sqrt{53.6} \approx 7.320$

Now, compute the two possible values of $x$:

$x_1 = \frac{-18 + 7.320}{4} = \frac{-10.68}{4} \approx -2.670$ $x_2 = \frac{-18 - 7.320}{4} = \frac{-25.32}{4} \approx -6.330$

Final result:

The zeros of the function $f(x) = 2x^2 + 18x + 33.8$ are approximately:

$x_1 \approx -2.670, \quad x_2 \approx -6.330$

Would you like more details or have any questions?

Here are 5 related questions:

1. What does the discriminant tell you about the nature of the roots?
2. How can you graphically interpret the zeros of a quadratic function?
3. How do you derive the quadratic formula from completing the square?
4. What happens if the discriminant is negative, and how would that affect the roots?
5. How can you factor a quadratic function, and is this one factorable?

Tip: The discriminant helps determine the number and type of solutions (real or complex).