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To solve these problems, we need to find the discriminant of each quadratic equation. The discriminant \( \Delta \) of a quadratic equation in the form \( ax^2 + bx + c = 0 \) is given by:

\[
\Delta = b^2 - 4ac
\]

Let's calculate the discriminant for each equation:

1. **Equation:** \( 4n^2 + 12n + 9 = 0 \)
   - \( a = 4 \), \( b = 12 \), \( c = 9 \)
   \[
   \Delta = 12^2 - 4(4)(9) = 144 - 144 = 0
   \]
   **Discriminant: 0**

2. **Equation:** \( 2x^2 - 3x + 1 = 0 \)
   - \( a = 2 \), \( b = -3 \), \( c = 1 \)
   \[
   \Delta = (-3)^2 - 4(2)(1) = 9 - 8 = 1
   \]
   **Discriminant: 1**

3. **Equation:** \( 4m^2 + 4m - 15 = 0 \)
   - \( a = 4 \), \( b = 4 \), \( c = -15 \)
   \[
   \Delta = 4^2 - 4(4)(-15) = 16 + 240 = 256
   \]
   **Discriminant: 256**

4. **Equation:** \( 4p^2 - p = 0 \)
   - Rewrite as \( 4p^2 - p + 0 = 0 \), so \( a = 4 \), \( b = -1 \), \( c = 0 \)
   \[
   \Delta = (-1)^2 - 4(4)(0) = 1 - 0 = 1
   \]
   **Discriminant: 1**

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Do you want more details or have any further questions? Here are five related questions that might help deepen your understanding:

1. What does the discriminant tell us about the number and type of solutions to a quadratic equation?
2. How do we find the roots of a quadratic equation when the discriminant is zero?
3. Can a quadratic equation have complex solutions? If so, how does the discriminant indicate this?
4. What does a positive discriminant signify in the context of quadratic equations?
5. How would you rewrite a quadratic equation in standard form if it's given in a different arrangement?

**Tip:** The discriminant is key to determining the nature of a quadratic equation's roots: if it's positive, there are two real roots; if zero, one real root; and if negative, the roots are complex.

Find the value of the discriminant in each quadratic equation: 4n² + 12n + 9 = 0, 2x² - 3x + 1 = 0, 4m² + 4m - 15 = 0, and 4p² - p = 0.

Learn how to calculate the discriminant for quadratic equations like 4n² + 12n + 9 = 0, 2x² - 3x + 1 = 0, and more. This problem involves the discriminant formula Δ = b² - 4ac, suitable for students in Grades 8-10.

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