Solve 21k^2-60k+32<0

$$21 k ^ { 2 } - 60 k + 32 < 0$$

Solve for k

$$k\in \left(-\frac{2\sqrt{57}}{21}+\frac{10}{7},\frac{2\sqrt{57}}{21}+\frac{10}{7}\right)$$

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.

$$21k^{2}-60k+32=0$$

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 21 for a, -60 for b, and 32 for c in the quadratic formula.

$$k=\frac{-\left(-60\right)±\sqrt{\left(-60\right)^{2}-4\times 21\times 32}}{2\times 21}$$

Do the calculations.

$$k=\frac{60±4\sqrt{57}}{42}$$

Solve the equation k=\frac{60±4\sqrt{57}}{42} when ± is plus and when ± is minus.

$$k=\frac{2\sqrt{57}}{21}+\frac{10}{7} k=-\frac{2\sqrt{57}}{21}+\frac{10}{7}$$

Rewrite the inequality by using the obtained solutions.

$$21\left(k-\left(\frac{2\sqrt{57}}{21}+\frac{10}{7}\right)\right)\left(k-\left(-\frac{2\sqrt{57}}{21}+\frac{10}{7}\right)\right)<0$$

For the product to be negative, k-\left(\frac{2\sqrt{57}}{21}+\frac{10}{7}\right) and k-\left(-\frac{2\sqrt{57}}{21}+\frac{10}{7}\right) have to be of the opposite signs. Consider the case when k-\left(\frac{2\sqrt{57}}{21}+\frac{10}{7}\right) is positive and k-\left(-\frac{2\sqrt{57}}{21}+\frac{10}{7}\right) is negative.

$$k-\left(\frac{2\sqrt{57}}{21}+\frac{10}{7}\right)>0 k-\left(-\frac{2\sqrt{57}}{21}+\frac{10}{7}\right)<0$$

This is false for any k.

$$k\in \emptyset$$

Consider the case when k-\left(-\frac{2\sqrt{57}}{21}+\frac{10}{7}\right) is positive and k-\left(\frac{2\sqrt{57}}{21}+\frac{10}{7}\right) is negative.

$$k-\left(-\frac{2\sqrt{57}}{21}+\frac{10}{7}\right)>0 k-\left(\frac{2\sqrt{57}}{21}+\frac{10}{7}\right)<0$$

The solution satisfying both inequalities is k\in \left(-\frac{2\sqrt{57}}{21}+\frac{10}{7},\frac{2\sqrt{57}}{21}+\frac{10}{7}\right).

$$k\in \left(-\frac{2\sqrt{57}}{21}+\frac{10}{7},\frac{2\sqrt{57}}{21}+\frac{10}{7}\right)$$

The final solution is the union of the obtained solutions.

$$k\in \left(-\frac{2\sqrt{57}}{21}+\frac{10}{7},\frac{2\sqrt{57}}{21}+\frac{10}{7}\right)$$