Solve 10n^2-383n+2079leq0

$$10 n ^ { 2 } - 383 n + 2079 \leq 0$$

Solve for n

$$n\in \begin{bmatrix}\frac{383-\sqrt{63529}}{20},\frac{\sqrt{63529}+383}{20}\end{bmatrix}$$

Steps Using the Quadratic Formula

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.

$$10n^{2}-383n+2079=0$$

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 10 for a, -383 for b, and 2079 for c in the quadratic formula.

$$n=\frac{-\left(-383\right)±\sqrt{\left(-383\right)^{2}-4\times 10\times 2079}}{2\times 10}$$

Do the calculations.

$$n=\frac{383±\sqrt{63529}}{20}$$

Solve the equation n=\frac{383±\sqrt{63529}}{20} when ± is plus and when ± is minus.

$$n=\frac{\sqrt{63529}+383}{20} n=\frac{383-\sqrt{63529}}{20}$$

Rewrite the inequality by using the obtained solutions.

$$10\left(n-\frac{\sqrt{63529}+383}{20}\right)\left(n-\frac{383-\sqrt{63529}}{20}\right)\leq 0$$

For the product to be ≤0, one of the values n-\frac{\sqrt{63529}+383}{20} and n-\frac{383-\sqrt{63529}}{20} has to be ≥0 and the other has to be ≤0. Consider the case when n-\frac{\sqrt{63529}+383}{20}\geq 0 and n-\frac{383-\sqrt{63529}}{20}\leq 0.

$$n-\frac{\sqrt{63529}+383}{20}\geq 0 n-\frac{383-\sqrt{63529}}{20}\leq 0$$

This is false for any n.

$$n\in \emptyset$$

Consider the case when n-\frac{\sqrt{63529}+383}{20}\leq 0 and n-\frac{383-\sqrt{63529}}{20}\geq 0.

$$n-\frac{383-\sqrt{63529}}{20}\geq 0 n-\frac{\sqrt{63529}+383}{20}\leq 0$$

The solution satisfying both inequalities is n\in \left[\frac{383-\sqrt{63529}}{20},\frac{\sqrt{63529}+383}{20}\right].

$$n\in \begin{bmatrix}\frac{383-\sqrt{63529}}{20},\frac{\sqrt{63529}+383}{20}\end{bmatrix}$$

The final solution is the union of the obtained solutions.

$$n\in \begin{bmatrix}\frac{383-\sqrt{63529}}{20},\frac{\sqrt{63529}+383}{20}\end{bmatrix}$$