The given equation is:

\[
8x^2 - 19x - 15 = 0
\]

This is a quadratic equation of the form \( ax^2 + bx + c = 0 \). We can solve it using the quadratic formula:

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

For your equation, \(a = 8\), \(b = -19\), and \(c = -15\). Plugging these values into the formula:

\[
x = \frac{-(-19) \pm \sqrt{(-19)^2 - 4(8)(-15)}}{2(8)}
\]

\[
x = \frac{19 \pm \sqrt{361 + 480}}{16}
\]

\[
x = \frac{19 \pm \sqrt{841}}{16}
\]

\[
x = \frac{19 \pm 29}{16}
\]

Now, we have two solutions:

1. \( x = \frac{19 + 29}{16} = \frac{48}{16} = 3 \)
2. \( x = \frac{19 - 29}{16} = \frac{-10}{16} = -\frac{5}{8} \)

Thus, the solutions are:

\[
x = 3 \quad \text{or} \quad x = -\frac{5}{8}
\]

Would you like further clarification on any step?

Here are some related questions:
1. How can you verify the solutions of a quadratic equation?
2. What are the steps to factor quadratic equations?
3. How do you apply the quadratic formula to real-life problems?
4. What happens when the discriminant (\(b^2 - 4ac\)) is negative?
5. How can graphing help solve quadratic equations?

**Tip:** Always check the discriminant (\(b^2 - 4ac\)) first to see how many real solutions exist.

Learn how to solve the quadratic equation 8x^2 - 19x - 15 using the quadratic formula. This step-by-step solution explores key algebraic concepts, such as identifying coefficients, calculating the discriminant, and finding roots. Suitable for students in Grades 8-10.

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