To solve the inequality $0 \leq N^2 + 152N - 4225$, we need to find the values of $N$ for which the quadratic expression is non-negative (i.e., greater than or equal to zero).

Steps:

1. Solve the equation $N^2 + 152N - 4225 = 0$: To find the critical points where the quadratic expression equals zero, we can apply the quadratic formula:

   $N = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

   For the equation $N^2 + 152N - 4225 = 0$, the coefficients are:

   • $a = 1$
   • $b = 152$
   • $c = -4225$

   Plugging these values into the quadratic formula:

   $N = \frac{-152 \pm \sqrt{152^2 - 4(1)(-4225)}}{2(1)}$

   Simplifying further:

   $N = \frac{-152 \pm \sqrt{23104 + 16900}}{2}$ $N = \frac{-152 \pm \sqrt{40004}}{2}$ $N = \frac{-152 \pm 200.01}{2}$

   Now we calculate the two possible solutions: $N_1 = \frac{-152 + 200.01}{2} = \frac{48.01}{2} \approx 24$ $N_2 = \frac{-152 - 200.01}{2} = \frac{-352.01}{2} \approx -176$

2. Analyze the sign of the quadratic: The roots of the quadratic are approximately $N = 24$ and $N = -176$. The quadratic expression $N^2 + 152N - 4225$ is a parabola that opens upwards (because the coefficient of $N^2$ is positive).

   The quadratic will be non-negative (i.e., $N^2 + 152N - 4225 \geq 0$) outside of the interval between the roots. So the solution to the inequality is:

   $N \leq -176 \quad \text{or} \quad N \geq 24$

3. Conclusion: The values of $N$ that satisfy the inequality are: $N \in (-\infty, -176] \cup [24, \infty)$

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

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Here are 5 related questions to expand on this:

1. How do you solve quadratic inequalities without using the quadratic formula?
2. Can you explain how to graph quadratic functions to visualize inequalities?
3. How does completing the square help solve quadratic inequalities?
4. What is the significance of the discriminant in solving quadratic inequalities?
5. How do you handle inequalities that involve absolute values?

Tip: Always check the sign of the leading coefficient when solving quadratic inequalities to know if the parabola opens upwards or downwards!