Solve -16t^2+64t+8

$$-16 { t }^{ 2 } +64t+8$$

Factor

$$-16\left(t-\left(-\frac{3\sqrt{2}}{2}+2\right)\right)\left(t-\left(\frac{3\sqrt{2}}{2}+2\right)\right)$$

Steps Using the Quadratic Formula

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.

$$-16t^{2}+64t+8=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$t=\frac{-64±\sqrt{64^{2}-4\left(-16\right)\times 8}}{2\left(-16\right)}$$

Square 64.

$$t=\frac{-64±\sqrt{4096-4\left(-16\right)\times 8}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$t=\frac{-64±\sqrt{4096+64\times 8}}{2\left(-16\right)}$$

Multiply 64 times 8.

$$t=\frac{-64±\sqrt{4096+512}}{2\left(-16\right)}$$

Add 4096 to 512.

$$t=\frac{-64±\sqrt{4608}}{2\left(-16\right)}$$

Take the square root of 4608.

$$t=\frac{-64±48\sqrt{2}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$t=\frac{-64±48\sqrt{2}}{-32}$$

Now solve the equation t=\frac{-64±48\sqrt{2}}{-32} when ± is plus. Add -64 to 48\sqrt{2}.

$$t=\frac{48\sqrt{2}-64}{-32}$$

Divide -64+48\sqrt{2} by -32.

$$t=-\frac{3\sqrt{2}}{2}+2$$

Now solve the equation t=\frac{-64±48\sqrt{2}}{-32} when ± is minus. Subtract 48\sqrt{2} from -64.

$$t=\frac{-48\sqrt{2}-64}{-32}$$

Divide -64-48\sqrt{2} by -32.

$$t=\frac{3\sqrt{2}}{2}+2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 2-\frac{3\sqrt{2}}{2} for x_{1} and 2+\frac{3\sqrt{2}}{2} for x_{2}.

$$-16t^{2}+64t+8=-16\left(t-\left(-\frac{3\sqrt{2}}{2}+2\right)\right)\left(t-\left(\frac{3\sqrt{2}}{2}+2\right)\right)$$

Evaluate

$$8+64t-16t^{2}$$