Solve -16t^2+80t+96=t

$$- 16 t ^ { 2 } + 80 t + 96 = t$$

Solve for t

$$t = \frac{\sqrt{12385} + 79}{32} \approx 5.94649734$$

$$t=\frac{79-\sqrt{12385}}{32}\approx -1.00899734$$

Steps Using the Quadratic Formula

Subtract t from both sides.

$$-16t^{2}+80t+96-t=0$$

Combine 80t and -t to get 79t.

$$-16t^{2}+79t+96=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 79 for b, and 96 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-79±\sqrt{79^{2}-4\left(-16\right)\times 96}}{2\left(-16\right)}$$

Square 79.

$$t=\frac{-79±\sqrt{6241-4\left(-16\right)\times 96}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$t=\frac{-79±\sqrt{6241+64\times 96}}{2\left(-16\right)}$$

Multiply 64 times 96.

$$t=\frac{-79±\sqrt{6241+6144}}{2\left(-16\right)}$$

Add 6241 to 6144.

$$t=\frac{-79±\sqrt{12385}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$t=\frac{-79±\sqrt{12385}}{-32}$$

Now solve the equation t=\frac{-79±\sqrt{12385}}{-32} when ± is plus. Add -79 to \sqrt{12385}.

$$t=\frac{\sqrt{12385}-79}{-32}$$

Divide -79+\sqrt{12385} by -32.

$$t=\frac{79-\sqrt{12385}}{32}$$

Now solve the equation t=\frac{-79±\sqrt{12385}}{-32} when ± is minus. Subtract \sqrt{12385} from -79.

$$t=\frac{-\sqrt{12385}-79}{-32}$$

Divide -79-\sqrt{12385} by -32.

$$t=\frac{\sqrt{12385}+79}{32}$$

The equation is now solved.

$$t=\frac{79-\sqrt{12385}}{32} t=\frac{\sqrt{12385}+79}{32}$$

Steps for Completing the Square

Subtract t from both sides.

$$-16t^{2}+80t+96-t=0$$

Combine 80t and -t to get 79t.

$$-16t^{2}+79t+96=0$$

Subtract 96 from both sides. Anything subtracted from zero gives its negation.

$$-16t^{2}+79t=-96$$

Divide both sides by -16.

$$\frac{-16t^{2}+79t}{-16}=-\frac{96}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$t^{2}+\frac{79}{-16}t=-\frac{96}{-16}$$

Divide 79 by -16.

$$t^{2}-\frac{79}{16}t=-\frac{96}{-16}$$

Divide -96 by -16.

$$t^{2}-\frac{79}{16}t=6$$

Divide -\frac{79}{16}, the coefficient of the x term, by 2 to get -\frac{79}{32}. Then add the square of -\frac{79}{32} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{79}{16}t+\left(-\frac{79}{32}\right)^{2}=6+\left(-\frac{79}{32}\right)^{2}$$

Square -\frac{79}{32} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{79}{16}t+\frac{6241}{1024}=6+\frac{6241}{1024}$$

Add 6 to \frac{6241}{1024}.

$$t^{2}-\frac{79}{16}t+\frac{6241}{1024}=\frac{12385}{1024}$$

Factor t^{2}-\frac{79}{16}t+\frac{6241}{1024}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{79}{32}\right)^{2}=\frac{12385}{1024}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{79}{32}\right)^{2}}=\sqrt{\frac{12385}{1024}}$$

Simplify.

$$t-\frac{79}{32}=\frac{\sqrt{12385}}{32} t-\frac{79}{32}=-\frac{\sqrt{12385}}{32}$$

Add \frac{79}{32} to both sides of the equation.

$$t=\frac{\sqrt{12385}+79}{32} t=\frac{79-\sqrt{12385}}{32}$$