The expression \(15x^2 + 10x - 7\) is a quadratic equation. Quadratic equations are polynomials of degree 2 and can be written in the general form:

\[
ax^2 + bx + c
\]

For the given equation:
- \(a = 15\)
- \(b = 10\)
- \(c = -7\)

### Steps to Solve the Quadratic Equation:
If you need to solve this quadratic equation (find the values of \(x\) where the expression equals zero), you can use the quadratic formula:

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

Substituting the values of \(a\), \(b\), and \(c\) into the formula:

\[
x = \frac{-10 \pm \sqrt{10^2 - 4(15)(-7)}}{2(15)}
\]

Let's simplify this step by step:

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

\[
\Delta = b^2 - 4ac = 10^2 - 4(15)(-7) = 100 + 420 = 520
\]

2. Substitute back into the quadratic formula:

\[
x = \frac{-10 \pm \sqrt{520}}{30}
\]

3. Simplify further. The square root of 520 can be expressed as \(\sqrt{520} = \sqrt{4 \times 130} = 2\sqrt{130}\):

\[
x = \frac{-10 \pm 2\sqrt{130}}{30} = \frac{-5 \pm \sqrt{130}}{15}
\]

### Final Answer:
The solutions to the equation are:

\[
x = \frac{-5 + \sqrt{130}}{15} \quad \text{and} \quad x = \frac{-5 - \sqrt{130}}{15}
\]

Would you like to go into more detail about any specific part of this solution? Here are some related questions:

1. How do you simplify square roots like \(\sqrt{520}\)?
2. What does the discriminant (\(\Delta\)) tell you about the nature of the roots?
3. Can this quadratic be factored easily instead of using the quadratic formula?
4. How can you approximate the decimal values of the roots?
5. What are the applications of solving quadratic equations in real-life problems?

**Tip:** When solving quadratic equations, always check if the discriminant (\(\Delta\)) is a perfect square, as it can simplify your calculations!

Learn how to solve the quadratic equation 15x^2 + 10x - 7 with a detailed step-by-step solution. This problem covers key algebraic concepts, including the quadratic formula and discriminant calculation. Suitable for students in Grades 10-12.

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