Solve 0.085q^2-11.9q+320leq0

$$0.085 q ^ { 2 } - 11.9 q + 320 \leq 0$$

Solve for q

$$q\in \begin{bmatrix}-\frac{10\sqrt{3281}}{17}+70,\frac{10\sqrt{3281}}{17}+70\end{bmatrix}$$

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.

$$0.085q^{2}-11.9q+320=0$$

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 0.085 for a, -11.9 for b, and 320 for c in the quadratic formula.

$$q=\frac{-\left(-11.9\right)±\sqrt{\left(-11.9\right)^{2}-4\times 0.085\times 320}}{0.085\times 2}$$

Do the calculations.

$$q=\frac{11.9±\frac{1}{10}\sqrt{3281}}{0.17}$$

Solve the equation q=\frac{11.9±\frac{1}{10}\sqrt{3281}}{0.17} when ± is plus and when ± is minus.

$$q=\frac{10\sqrt{3281}}{17}+70 q=-\frac{10\sqrt{3281}}{17}+70$$

Rewrite the inequality by using the obtained solutions.

$$0.085\left(q-\left(\frac{10\sqrt{3281}}{17}+70\right)\right)\left(q-\left(-\frac{10\sqrt{3281}}{17}+70\right)\right)\leq 0$$

For the product to be ≤0, one of the values q-\left(\frac{10\sqrt{3281}}{17}+70\right) and q-\left(-\frac{10\sqrt{3281}}{17}+70\right) has to be ≥0 and the other has to be ≤0. Consider the case when q-\left(\frac{10\sqrt{3281}}{17}+70\right)\geq 0 and q-\left(-\frac{10\sqrt{3281}}{17}+70\right)\leq 0.

$$q-\left(\frac{10\sqrt{3281}}{17}+70\right)\geq 0 q-\left(-\frac{10\sqrt{3281}}{17}+70\right)\leq 0$$

This is false for any q.

$$q\in \emptyset$$

Consider the case when q-\left(\frac{10\sqrt{3281}}{17}+70\right)\leq 0 and q-\left(-\frac{10\sqrt{3281}}{17}+70\right)\geq 0.

$$q-\left(-\frac{10\sqrt{3281}}{17}+70\right)\geq 0 q-\left(\frac{10\sqrt{3281}}{17}+70\right)\leq 0$$

The solution satisfying both inequalities is q\in \left[-\frac{10\sqrt{3281}}{17}+70,\frac{10\sqrt{3281}}{17}+70\right].

$$q\in \begin{bmatrix}-\frac{10\sqrt{3281}}{17}+70,\frac{10\sqrt{3281}}{17}+70\end{bmatrix}$$

The final solution is the union of the obtained solutions.

$$q\in \begin{bmatrix}-\frac{10\sqrt{3281}}{17}+70,\frac{10\sqrt{3281}}{17}+70\end{bmatrix}$$