Solve -16t^2+40t+4

$$-16 { t }^{ 2 } +40t+4$$

Factor

$$-16\left(t-\frac{5-\sqrt{29}}{4}\right)\left(t-\frac{\sqrt{29}+5}{4}\right)$$

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.

$$-16t^{2}+40t+4=0$$

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.

$$t=\frac{-40±\sqrt{40^{2}-4\left(-16\right)\times 4}}{2\left(-16\right)}$$

Square 40.

$$t=\frac{-40±\sqrt{1600-4\left(-16\right)\times 4}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$t=\frac{-40±\sqrt{1600+64\times 4}}{2\left(-16\right)}$$

Multiply 64 times 4.

$$t=\frac{-40±\sqrt{1600+256}}{2\left(-16\right)}$$

Add 1600 to 256.

$$t=\frac{-40±\sqrt{1856}}{2\left(-16\right)}$$

Take the square root of 1856.

$$t=\frac{-40±8\sqrt{29}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$t=\frac{-40±8\sqrt{29}}{-32}$$

Now solve the equation t=\frac{-40±8\sqrt{29}}{-32} when ± is plus. Add -40 to 8\sqrt{29}.

$$t=\frac{8\sqrt{29}-40}{-32}$$

Divide -40+8\sqrt{29} by -32.

$$t=\frac{5-\sqrt{29}}{4}$$

Now solve the equation t=\frac{-40±8\sqrt{29}}{-32} when ± is minus. Subtract 8\sqrt{29} from -40.

$$t=\frac{-8\sqrt{29}-40}{-32}$$

Divide -40-8\sqrt{29} by -32.

$$t=\frac{\sqrt{29}+5}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5-\sqrt{29}}{4} for x_{1} and \frac{5+\sqrt{29}}{4} for x_{2}.

$$-16t^{2}+40t+4=-16\left(t-\frac{5-\sqrt{29}}{4}\right)\left(t-\frac{\sqrt{29}+5}{4}\right)$$

Evaluate

$$4+40t-16t^{2}$$