You have the quadratic expression \(5x^2 + 19x + 12\). To solve it (if you're looking for the roots), you can use the **quadratic formula**:

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

Here, \(a = 5\), \(b = 19\), and \(c = 12\).

Let's solve it step by step:

1. **Discriminant** (\( \Delta \)) calculation:
   \[
   \Delta = b^2 - 4ac = 19^2 - 4(5)(12) = 361 - 240 = 121
   \]

2. **Quadratic formula application**:
   \[
   x = \frac{-19 \pm \sqrt{121}}{2(5)} = \frac{-19 \pm 11}{10}
   \]

3. Solving for the two possible values of \(x\):

   - For \( x_1 \):
     \[
     x_1 = \frac{-19 + 11}{10} = \frac{-8}{10} = -0.8
     \]

   - For \( x_2 \):
     \[
     x_2 = \frac{-19 - 11}{10} = \frac{-30}{10} = -3
     \]

The solutions are:
\[
x = -0.8 \quad \text{or} \quad x = -3
\]

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

Here are 5 related questions:
1. How do you determine if a quadratic equation has real or complex roots?
2. Can you solve this quadratic by factoring?
3. How do you complete the square to solve a quadratic equation?
4. What is the significance of the discriminant in a quadratic equation?
5. How does the quadratic formula apply to equations with imaginary roots?

**Tip:** The discriminant (\(b^2 - 4ac\)) tells you whether the roots are real and distinct, real and repeated, or complex!

Discover how to solve the quadratic equation 5x^2 + 19x + 12 step-by-step using the quadratic formula. This guide explains key algebraic concepts such as the discriminant and provides detailed calculations. Suitable for Grades 9-10 students.

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