Solve 256*3t^2+640*2t+396>0

$$256 \times 3 t ^ { 2 } + 640 \times 2 t + 396 > 0$$

Solve for t

$$t\in \left(-\infty,-\frac{\sqrt{103}}{24}-\frac{5}{6}\right)\cup \left(\frac{\sqrt{103}}{24}-\frac{5}{6},\infty\right)$$

Steps Using the Quadratic Formula

Do the multiplications.

$$768t^{2}+1280t+396>0$$

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.

$$768t^{2}+1280t+396=0$$

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 768 for a, 1280 for b, and 396 for c in the quadratic formula.

$$t=\frac{-1280±\sqrt{1280^{2}-4\times 768\times 396}}{2\times 768}$$

Do the calculations.

$$t=\frac{-1280±64\sqrt{103}}{1536}$$

Solve the equation t=\frac{-1280±64\sqrt{103}}{1536} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{103}}{24}-\frac{5}{6} t=-\frac{\sqrt{103}}{24}-\frac{5}{6}$$

Rewrite the inequality by using the obtained solutions.

$$768\left(t-\left(\frac{\sqrt{103}}{24}-\frac{5}{6}\right)\right)\left(t-\left(-\frac{\sqrt{103}}{24}-\frac{5}{6}\right)\right)>0$$

For the product to be positive, t-\left(\frac{\sqrt{103}}{24}-\frac{5}{6}\right) and t-\left(-\frac{\sqrt{103}}{24}-\frac{5}{6}\right) have to be both negative or both positive. Consider the case when t-\left(\frac{\sqrt{103}}{24}-\frac{5}{6}\right) and t-\left(-\frac{\sqrt{103}}{24}-\frac{5}{6}\right) are both negative.

$$t-\left(\frac{\sqrt{103}}{24}-\frac{5}{6}\right)<0 t-\left(-\frac{\sqrt{103}}{24}-\frac{5}{6}\right)<0$$

The solution satisfying both inequalities is t<-\frac{\sqrt{103}}{24}-\frac{5}{6}.

$$t<-\frac{\sqrt{103}}{24}-\frac{5}{6}$$

Consider the case when t-\left(\frac{\sqrt{103}}{24}-\frac{5}{6}\right) and t-\left(-\frac{\sqrt{103}}{24}-\frac{5}{6}\right) are both positive.

$$t-\left(-\frac{\sqrt{103}}{24}-\frac{5}{6}\right)>0 t-\left(\frac{\sqrt{103}}{24}-\frac{5}{6}\right)>0$$

The solution satisfying both inequalities is t>\frac{\sqrt{103}}{24}-\frac{5}{6}.

$$t>\frac{\sqrt{103}}{24}-\frac{5}{6}$$

The final solution is the union of the obtained solutions.

$$t<-\frac{\sqrt{103}}{24}-\frac{5}{6}\text{; }t>\frac{\sqrt{103}}{24}-\frac{5}{6}$$