https://file.aiquickdraw.com/mathbot/file/1727272022229yglb9dtn.jpg

The function given in the image is:

\[
f(x) = \frac{(x - 3)(-6x^2 + 17x - 5)}{x^2}
\]

We need to determine the zeros of this function.

### Step 1: Finding the zeros of the numerator
The function is zero when the numerator \((x - 3)(-6x^2 + 17x - 5)\) equals zero. So, we solve:

\[
(x - 3)(-6x^2 + 17x - 5) = 0
\]

This gives us two cases:
1. \(x - 3 = 0\) implies \(x = 3\).
2. \(-6x^2 + 17x - 5 = 0\).

### Step 2: Solving the quadratic equation
Now, solve the quadratic equation \(-6x^2 + 17x - 5 = 0\) using the quadratic formula:

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

For the equation \(-6x^2 + 17x - 5 = 0\), the coefficients are \(a = -6\), \(b = 17\), and \(c = -5\). Plugging these into the quadratic formula:

\[
x = \frac{-17 \pm \sqrt{17^2 - 4(-6)(-5)}}{2(-6)}
\]
\[
x = \frac{-17 \pm \sqrt{289 - 120}}{-12}
\]
\[
x = \frac{-17 \pm \sqrt{169}}{-12}
\]
\[
x = \frac{-17 \pm 13}{-12}
\]

This gives two solutions:
1. \(x = \frac{-17 + 13}{-12} = \frac{-4}{-12} = \frac{1}{3}\),
2. \(x = \frac{-17 - 13}{-12} = \frac{-30}{-12} = \frac{5}{2}\).

### Step 3: Checking the zeros of the denominator
The denominator of the function is \(x^2\), which is zero when \(x = 0\). However, this is not a zero of the function but a point of discontinuity because division by zero is undefined.

### Final Answer
The zeros of the function are \(x = 3\), \(x = \frac{1}{3}\), and \(x = \frac{5}{2}\).

The correct answer is the option with \( \frac{1}{3}, \frac{5}{2}, \) and \( 3 \). This corresponds to the first option.

Would you like more details or have any questions? Here are five related questions to expand your understanding:

1. How does the quadratic formula work, and why is it useful?
2. What are the conditions that cause a function to have discontinuities?
3. How does the factorization of a function help in finding its zeros?
4. Can you explain how to handle cases where the denominator introduces points of discontinuity?
5. How can we check if a zero is a root or a point of discontinuity in a rational function?

**Tip**: When solving quadratic equations, always double-check the discriminant (\(b^2 - 4ac\)) to determine if there are real solutions.

The zeros of the function f(x) = \frac{(x - 3)(-6x^2 + 17x - 5)}{x^2} are?

Learn how to find the zeros of a rational function f(x) = (x - 3)(-6x^2 + 17x - 5) / x^2 by solving its numerator using the quadratic formula and linear factor. This guide covers key algebraic concepts for high school students in Grades 9-11.

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