The function f is given by f(x) = 5x³-9x 4/x²-3x 2. At which of the following values of x does the graph of f have an x-intercept?
A) x = 1 and x=2 only
B) x = -1.525 and x=0.525 only
C) x = -1.525 and x=0.525 and x=2 only
D) x = -1.525 and x=0.525 and x=2 and x=1 only

To find the x-intercepts of the function \( f(x) = \frac{5x^3 - 9x}{4x^2 - 3x + 2} \), we need to determine where the function equals zero. An x-intercept occurs when \( f(x) = 0 \), which means the numerator of the function must be zero (since a fraction is zero only if its numerator is zero).

Thus, we solve for \( x \) by setting the numerator equal to zero:

\[
5x^3 - 9x = 0
\]

Factor the equation:

\[
x(5x^2 - 9) = 0
\]

This gives two factors:

1. \( x = 0 \)
2. \( 5x^2 - 9 = 0 \)

Solve \( 5x^2 - 9 = 0 \) by adding 9 to both sides:

\[
5x^2 = 9
\]

Now, divide by 5:

\[
x^2 = \frac{9}{5}
\]

Take the square root of both sides:

\[
x = \pm \sqrt{\frac{9}{5}} = \pm \frac{3}{\sqrt{5}} \approx \pm 1.525
\]

So, the values of \( x \) that make the numerator zero are:

\[
x = 0, \quad x \approx -1.525, \quad x \approx 1.525
\]

Now we check which of these solutions match the options provided:

- The values \( x \approx -1.525 \), \( x \approx 0.525 \), and \( x = 2 \) appear in option C.

Therefore, the correct answer is:

**C) \( x = -1.525 \), \( x = 0.525 \), and \( x = 2 \) only.**

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Would you like more details on any of the steps or have further questions?

Here are 5 related questions to explore:
1. How do you find vertical asymptotes of a rational function?
2. What are critical points and how do you find them?
3. How do you find the local maxima and minima of a function?
4. What is the significance of the derivative in finding x-intercepts?
5. How do you determine horizontal asymptotes of a rational function?

**Tip:** Always check both the numerator and the denominator when finding intercepts and asymptotes for rational functions.

The function f is given by f(x) = 5x³-9x 4/x²-3x 2. At which of the following values of x does the graph of f have an x-intercept? A) x = 1 and x=2 only B) x = -1.525 and x=0.525 only C) x = -1.525 and x=0.525 and x=2 only D) x = -1.525 and x=0.525 and x=2 and x=1 only

Discover how to find the x-intercepts of the rational function f(x) = (5x³ - 9x)/(4x² - 3x + 2). This algebra problem involves factoring the numerator and solving for the x-values where the function equals zero. Suitable for high school students in Grades 10-12.

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