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Let's solve the problem step-by-step:

Given functions:
- \( f(x) = 3x^2 + 2x - 5 \)
- \( g(x) = 2x - 1 \)

We need to find the value of \( x \) such that \( f(g(x)) = 0 \).

### Step 1: Substituting \( g(x) \) into \( f(x) \)

We are asked to find \( f(g(x)) \). So, substitute \( g(x) = 2x - 1 \) into \( f(x) \):

\[
f(g(x)) = 3(g(x))^2 + 2(g(x)) - 5
\]

Substitute \( g(x) = 2x - 1 \):

\[
f(2x - 1) = 3(2x - 1)^2 + 2(2x - 1) - 5
\]

### Step 2: Expanding the expression

Expand \( (2x - 1)^2 \):

\[
(2x - 1)^2 = 4x^2 - 4x + 1
\]

Now substitute this back into the equation:

\[
f(2x - 1) = 3(4x^2 - 4x + 1) + 2(2x - 1) - 5
\]

Expand further:

\[
f(2x - 1) = 12x^2 - 12x + 3 + 4x - 2 - 5
\]

Simplify:

\[
f(2x - 1) = 12x^2 - 8x - 4
\]

### Step 3: Solve \( f(g(x)) = 0 \)

Now solve \( 12x^2 - 8x - 4 = 0 \). Divide the entire equation by 4 to simplify:

\[
3x^2 - 2x - 1 = 0
\]

This is a quadratic equation. We can solve it using the quadratic formula:

\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(3)(-1)}}{2(3)}
\]

\[
x = \frac{2 \pm \sqrt{4 + 12}}{6}
\]

\[
x = \frac{2 \pm \sqrt{16}}{6}
\]

\[
x = \frac{2 \pm 4}{6}
\]

So the two solutions are:

\[
x = \frac{2 + 4}{6} = 1 \quad \text{or} \quad x = \frac{2 - 4}{6} = -\frac{1}{3}
\]

### Final Answer:
The values of \( x \) that satisfy \( f(g(x)) = 0 \) are \( x = 1 \) and \( x = -\frac{1}{3} \).

Would you like further details on any step?

Here are 5 related questions to explore:

1. How do we derive the quadratic formula?
2. Can you explain how to expand binomials like \( (2x - 1)^2 \)?
3. What other methods can solve quadratic equations besides the quadratic formula?
4. How do you verify that both solutions \( x = 1 \) and \( x = -\frac{1}{3} \) satisfy \( f(g(x)) = 0 \)?
5. What role does the discriminant play in determining the nature of solutions?

**Tip:** Always simplify quadratic equations by factoring or dividing when possible before using the quadratic formula—it can make the calculations easier.

If f(x) = 3x^2 + 2x - 5 and g(x) = 2x - 1, for what value of x does f(g(x)) = 0?

Learn how to solve the composite function f(g(x)) = 0 where f(x) = 3x^2 + 2x - 5 and g(x) = 2x - 1. This problem covers key algebraic concepts such as composite functions, quadratic equations, and the quadratic formula. Suitable for students in Grades 9-12.

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