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Let's analyze the problem step by step.

We are given the function \( f(t) = t^2 + 3t - 1 \), which describes the position of a moving object over time. The task is to determine at what time \( t \) the object is at position 9, meaning we need to solve the equation:

\[
f(t) = 9
\]

Substituting \( f(t) = t^2 + 3t - 1 \) into the equation, we get:

\[
t^2 + 3t - 1 = 9
\]

Now, solve this equation:

\[
t^2 + 3t - 1 - 9 = 0
\]

\[
t^2 + 3t - 10 = 0
\]

We now solve the quadratic equation \( t^2 + 3t - 10 = 0 \) using the quadratic formula:

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

For the equation \( t^2 + 3t - 10 = 0 \), we have:
- \( a = 1 \)
- \( b = 3 \)
- \( c = -10 \)

Substitute these values into the quadratic formula:

\[
t = \frac{-3 \pm \sqrt{3^2 - 4(1)(-10)}}{2(1)}
\]

\[
t = \frac{-3 \pm \sqrt{9 + 40}}{2}
\]

\[
t = \frac{-3 \pm \sqrt{49}}{2}
\]

\[
t = \frac{-3 \pm 7}{2}
\]

So, we have two possible solutions:

\[
t_1 = \frac{-3 + 7}{2} = \frac{4}{2} = 2
\]
\[
t_2 = \frac{-3 - 7}{2} = \frac{-10}{2} = -5
\]

Since negative time doesn't make sense in this context, the valid solution is \( t = 2 \).

Thus, the correct answer is **A) \( t = 2 \)**.

Would you like further details or have any questions about this?

Here are 5 related questions to deepen understanding:
1. How would the answer change if the position was different from 9?
2. What does the derivative of \( f(t) = t^2 + 3t - 1 \) tell us about the motion of the object?
3. Could we solve this using graphical methods, and how would the graph of \( f(t) \) help visualize the solution?
4. What is the significance of the second solution \( t = -5 \), and why do we discard it?
5. How can the discriminant of a quadratic equation help us predict the number of real solutions?

**Tip:** When solving quadratic equations, always check if factoring is possible before using the quadratic formula—it can save time!

Considere f(t) = t^2 + 3t - 1 a função que corresponde ao trajeto de um objeto móvel de acordo com o tempo. Em qual instante o móvel se encontra na posição 9?

Learn how to solve a quadratic equation where f(t) = t^2 + 3t - 1 describes the position of a moving object over time. The problem involves finding the time when the object is at position 9. This is a detailed solution using the quadratic formula, suitable for students in Grades 9-11.

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