Solve 7n^2-121n-128-3728geq0

$$7 n ^ { 2 } - 121 n - 128 - 3728 \geq 0$$

Solve for n

$$n\in (-\infty,\frac{121-\sqrt{122609}}{14}]\cup [\frac{\sqrt{122609}+121}{14},\infty)$$

Steps Using the Quadratic Formula

Subtract 3728 from -128 to get -3856.

$$7n^{2}-121n-3856\geq 0$$

To solve the inequality, factor the left hand side. Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$7n^{2}-121n-3856=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 7 for a, -121 for b, and -3856 for c in the quadratic formula.

$$n=\frac{-\left(-121\right)±\sqrt{\left(-121\right)^{2}-4\times 7\left(-3856\right)}}{2\times 7}$$

Do the calculations.

$$n=\frac{121±\sqrt{122609}}{14}$$

Solve the equation n=\frac{121±\sqrt{122609}}{14} when ± is plus and when ± is minus.

$$n=\frac{\sqrt{122609}+121}{14} n=\frac{121-\sqrt{122609}}{14}$$

Rewrite the inequality by using the obtained solutions.

$$7\left(n-\frac{\sqrt{122609}+121}{14}\right)\left(n-\frac{121-\sqrt{122609}}{14}\right)\geq 0$$

For the product to be ≥0, n-\frac{\sqrt{122609}+121}{14} and n-\frac{121-\sqrt{122609}}{14} have to be both ≤0 or both ≥0. Consider the case when n-\frac{\sqrt{122609}+121}{14} and n-\frac{121-\sqrt{122609}}{14} are both ≤0.

$$n-\frac{\sqrt{122609}+121}{14}\leq 0 n-\frac{121-\sqrt{122609}}{14}\leq 0$$

The solution satisfying both inequalities is n\leq \frac{121-\sqrt{122609}}{14}.

$$n\leq \frac{121-\sqrt{122609}}{14}$$

Consider the case when n-\frac{\sqrt{122609}+121}{14} and n-\frac{121-\sqrt{122609}}{14} are both ≥0.

$$n-\frac{121-\sqrt{122609}}{14}\geq 0 n-\frac{\sqrt{122609}+121}{14}\geq 0$$

The solution satisfying both inequalities is n\geq \frac{\sqrt{122609}+121}{14}.

$$n\geq \frac{\sqrt{122609}+121}{14}$$

The final solution is the union of the obtained solutions.

$$n\leq \frac{121-\sqrt{122609}}{14}\text{; }n\geq \frac{\sqrt{122609}+121}{14}$$