The expression you've provided, \(10x^2 + 3x - 4\), is a quadratic expression. Here’s a breakdown of its components:

- **Coefficient of \(x^2\)**: 10
- **Coefficient of \(x\)**: 3
- **Constant term**: -4

If you want to solve this quadratic expression (i.e., find the roots where the expression equals zero), you can use the quadratic formula:

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

Where:
- \( a = 10 \) (coefficient of \(x^2\))
- \( b = 3 \) (coefficient of \(x\))
- \( c = -4 \) (constant term)

Let's apply this formula:

1. **Calculate the discriminant**: 
   \[
   \text{Discriminant} = b^2 - 4ac = 3^2 - 4(10)(-4) = 9 + 160 = 169
   \]
   
2. **Square root of the discriminant**: 
   \[
   \sqrt{169} = 13
   \]
   
3. **Apply the quadratic formula**:
   \[
   x = \frac{-3 \pm 13}{2(10)}
   \]

   This gives us two possible solutions:
   \[
   x_1 = \frac{-3 + 13}{20} = \frac{10}{20} = 0.5
   \]
   \[
   x_2 = \frac{-3 - 13}{20} = \frac{-16}{20} = -0.8
   \]

So, the solutions to the quadratic equation \(10x^2 + 3x - 4 = 0\) are \(x = 0.5\) and \(x = -0.8\).

Would you like more details on any step, or do you have any questions?

### Related Questions:
1. What happens if the discriminant is negative?
2. Can we factor this quadratic equation directly instead of using the quadratic formula?
3. How would the graph of \(10x^2 + 3x - 4\) look?
4. What is the vertex of the quadratic equation?
5. How can we use completing the square to solve this quadratic equation?

**Tip:** The discriminant (\(b^2 - 4ac\)) tells you the nature of the roots: if it's positive, the equation has two real roots; if it's zero, one real root; and if it's negative, the roots are complex.

Learn how to solve the quadratic equation 10x^2 + 3x - 4 = 0 step-by-step using the quadratic formula. Understand the discriminant and find the roots of the equation. Suitable for high school students and advanced learners.

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