To solve the quadratic equation \(16t^2 + 15t - 171 = 0\), we can use the quadratic formula:

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

Here, \(a = 16\), \(b = 15\), and \(c = -171\). Let's plug these values into the quadratic formula:

1. Calculate the discriminant (\(\Delta\)):

\[
\Delta = b^2 - 4ac = 15^2 - 4 \cdot 16 \cdot (-171)
\]

2. Simplify the discriminant:

\[
\Delta = 225 + 2736 = 2961
\]

3. Apply the quadratic formula:

\[
t = \frac{-15 \pm \sqrt{2961}}{2 \cdot 16}
\]

4. Simplify further:

\[
t = \frac{-15 \pm \sqrt{2961}}{32}
\]

Thus, the solutions are:

\[
t = \frac{-15 + \sqrt{2961}}{32} \quad \text{and} \quad t = \frac{-15 - \sqrt{2961}}{32}
\]

Would you like further details or have any questions?

Here are some related questions:

1. How do you find the roots of a quadratic equation using the quadratic formula?
2. What is the discriminant, and how does it determine the nature of the roots?
3. How do you simplify square roots in quadratic equations?
4. What are the steps for solving quadratic equations by factoring?
5. How do you complete the square to solve quadratic equations?
6. What is the significance of the coefficients in a quadratic equation?
7. How can you use graphing to find the solutions of a quadratic equation?
8. What are some real-world applications of quadratic equations?

**Tip:** Always check your calculations when applying the quadratic formula to ensure accuracy.

Learn how to solve the quadratic equation 16t^2 + 15t - 171 with a detailed step-by-step solution using the quadratic formula. This problem covers key algebraic concepts, including the discriminant and quadratic formula. Suitable for high school students in Grades 10-12.

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