Solve -16t^2+112t+30

$$-16 { t }^{ 2 } +112t+30$$

Factor

$$-16\left(t-\left(-\frac{\sqrt{226}}{4}+\frac{7}{2}\right)\right)\left(t-\left(\frac{\sqrt{226}}{4}+\frac{7}{2}\right)\right)$$

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.

$$-16t^{2}+112t+30=0$$

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.

$$t=\frac{-112±\sqrt{112^{2}-4\left(-16\right)\times 30}}{2\left(-16\right)}$$

Square 112.

$$t=\frac{-112±\sqrt{12544-4\left(-16\right)\times 30}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$t=\frac{-112±\sqrt{12544+64\times 30}}{2\left(-16\right)}$$

Multiply 64 times 30.

$$t=\frac{-112±\sqrt{12544+1920}}{2\left(-16\right)}$$

Add 12544 to 1920.

$$t=\frac{-112±\sqrt{14464}}{2\left(-16\right)}$$

Take the square root of 14464.

$$t=\frac{-112±8\sqrt{226}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$t=\frac{-112±8\sqrt{226}}{-32}$$

Now solve the equation t=\frac{-112±8\sqrt{226}}{-32} when ± is plus. Add -112 to 8\sqrt{226}.

$$t=\frac{8\sqrt{226}-112}{-32}$$

Divide -112+8\sqrt{226} by -32.

$$t=-\frac{\sqrt{226}}{4}+\frac{7}{2}$$

Now solve the equation t=\frac{-112±8\sqrt{226}}{-32} when ± is minus. Subtract 8\sqrt{226} from -112.

$$t=\frac{-8\sqrt{226}-112}{-32}$$

Divide -112-8\sqrt{226} by -32.

$$t=\frac{\sqrt{226}}{4}+\frac{7}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7}{2}-\frac{\sqrt{226}}{4} for x_{1} and \frac{7}{2}+\frac{\sqrt{226}}{4} for x_{2}.

$$-16t^{2}+112t+30=-16\left(t-\left(-\frac{\sqrt{226}}{4}+\frac{7}{2}\right)\right)\left(t-\left(\frac{\sqrt{226}}{4}+\frac{7}{2}\right)\right)$$

Evaluate

$$30+112t-16t^{2}$$