3n^2 - 100 n + 6 > c.n for a certain value n
how do i find this value n

To find the value \( n \) such that the inequality \( 3n^2 - 100n + 6 > c \cdot n \) holds, we need to solve for \( n \) in terms of the constant \( c \). Follow these steps:

1. **Rewrite the inequality:**
   \[
   3n^2 - 100n + 6 > c \cdot n
   \]
   
2. **Move all terms to one side:**
   \[
   3n^2 - 100n + 6 - c \cdot n > 0
   \]
   \[
   3n^2 - (100 + c)n + 6 > 0
   \]

3. **Solve the corresponding quadratic equation:**
   We solve the quadratic equation \( 3n^2 - (100 + c)n + 6 = 0 \) to find the critical points. The general form of a quadratic equation is \( ax^2 + bx + c = 0 \), where:
   \[
   a = 3, \quad b = -(100 + c), \quad c = 6
   \]

4. **Calculate the discriminant:**
   \[
   \Delta = b^2 - 4ac
   \]
   \[
   \Delta = (-(100 + c))^2 - 4 \cdot 3 \cdot 6
   \]
   \[
   \Delta = (100 + c)^2 - 72
   \]

5. **Find the roots:**
   The roots \( n_1 \) and \( n_2 \) are given by the quadratic formula:
   \[
   n = \frac{-b \pm \sqrt{\Delta}}{2a}
   \]
   \[
   n = \frac{100 + c \pm \sqrt{(100 + c)^2 - 72}}{6}
   \]

6. **Determine the intervals:**
   The quadratic equation \( 3n^2 - (100 + c)n + 6 = 0 \) will have two roots, say \( n_1 \) and \( n_2 \), where \( n_1 \le n_2 \). The quadratic expression \( 3n^2 - (100 + c)n + 6 \) will be positive outside the interval \((n_1, n_2)\) and negative inside the interval \((n_1, n_2)\).

Thus, the value of \( n \) that satisfies the inequality \( 3n^2 - 100n + 6 > c \cdot n \) will be either:
\[
n < n_1 \quad \text{or} \quad n > n_2
\]

Would you like further details on any of these steps or have any questions?

Here are five questions you might find helpful to ask next:
1. Can you show me an example with specific values for \( c \)?
2. How do I calculate the discriminant step by step?
3. What is the significance of the discriminant in solving quadratic equations?
4. How do I graphically interpret the solution to this inequality?
5. Can you explain the quadratic formula in more detail?

**Tip:** When solving quadratic inequalities, it's helpful to first find the roots of the corresponding quadratic equation and then determine the sign of the quadratic expression in the intervals defined by these roots.

Learn how to solve the quadratic inequality 3n^2 - 100n + 6 > c.n with a detailed step-by-step solution. This problem covers key algebraic concepts, including quadratic equations, inequalities, and the discriminant formula. Suitable for advanced high school students and beyond.

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