Solve 500/3969x^2-2500/3969*x-16720/3969leq0

$$\frac { 500 } { 3969 } x ^ { 2 } - \frac { 2500 } { 3969 } \cdot x - \frac { 16720 } { 3969 } \leq 0$$

Solve for x

$$x\in \begin{bmatrix}-\frac{19}{5},\frac{44}{5}\end{bmatrix}$$

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.

$$\frac{500}{3969}x^{2}-\frac{2500}{3969}x-\frac{16720}{3969}=0$$

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 \frac{500}{3969} for a, -\frac{2500}{3969} for b, and -\frac{16720}{3969} for c in the quadratic formula.

$$x=\frac{-\left(-\frac{2500}{3969}\right)±\sqrt{\left(-\frac{2500}{3969}\right)^{2}-4\times \frac{500}{3969}\left(-\frac{16720}{3969}\right)}}{\frac{500}{3969}\times 2}$$

Do the calculations.

$$x=\frac{\frac{2500}{3969}±\frac{100}{63}}{\frac{1000}{3969}}$$

Solve the equation x=\frac{\frac{2500}{3969}±\frac{100}{63}}{\frac{1000}{3969}} when ± is plus and when ± is minus.

$$x=\frac{44}{5} x=-\frac{19}{5}$$

Rewrite the inequality by using the obtained solutions.

$$\frac{500}{3969}\left(x-\frac{44}{5}\right)\left(x+\frac{19}{5}\right)\leq 0$$

For the product to be ≤0, one of the values x-\frac{44}{5} and x+\frac{19}{5} has to be ≥0 and the other has to be ≤0. Consider the case when x-\frac{44}{5}\geq 0 and x+\frac{19}{5}\leq 0.

$$x-\frac{44}{5}\geq 0 x+\frac{19}{5}\leq 0$$

This is false for any x.

$$x\in \emptyset$$

Consider the case when x-\frac{44}{5}\leq 0 and x+\frac{19}{5}\geq 0.

$$x+\frac{19}{5}\geq 0 x-\frac{44}{5}\leq 0$$

The solution satisfying both inequalities is x\in \left[-\frac{19}{5},\frac{44}{5}\right].

$$x\in \begin{bmatrix}-\frac{19}{5},\frac{44}{5}\end{bmatrix}$$

The final solution is the union of the obtained solutions.

$$x\in \begin{bmatrix}-\frac{19}{5},\frac{44}{5}\end{bmatrix}$$