Solve 60=-16t^2+70t+5

$$60 = - 16 t ^ { 2 } + 70 t + 5$$

Solve for t

$$t = \frac{\sqrt{345} + 35}{16} \approx 3.348385976$$

$$t = \frac{35 - \sqrt{345}}{16} \approx 1.026614024$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$-16t^{2}+70t+5=60$$

Subtract 60 from both sides.

$$-16t^{2}+70t+5-60=0$$

Subtract 60 from 5 to get -55.

$$-16t^{2}+70t-55=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 70 for b, and -55 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-70±\sqrt{70^{2}-4\left(-16\right)\left(-55\right)}}{2\left(-16\right)}$$

Square 70.

$$t=\frac{-70±\sqrt{4900-4\left(-16\right)\left(-55\right)}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$t=\frac{-70±\sqrt{4900+64\left(-55\right)}}{2\left(-16\right)}$$

Multiply 64 times -55.

$$t=\frac{-70±\sqrt{4900-3520}}{2\left(-16\right)}$$

Add 4900 to -3520.

$$t=\frac{-70±\sqrt{1380}}{2\left(-16\right)}$$

Take the square root of 1380.

$$t=\frac{-70±2\sqrt{345}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$t=\frac{-70±2\sqrt{345}}{-32}$$

Now solve the equation t=\frac{-70±2\sqrt{345}}{-32} when ± is plus. Add -70 to 2\sqrt{345}.

$$t=\frac{2\sqrt{345}-70}{-32}$$

Divide -70+2\sqrt{345} by -32.

$$t=\frac{35-\sqrt{345}}{16}$$

Now solve the equation t=\frac{-70±2\sqrt{345}}{-32} when ± is minus. Subtract 2\sqrt{345} from -70.

$$t=\frac{-2\sqrt{345}-70}{-32}$$

Divide -70-2\sqrt{345} by -32.

$$t=\frac{\sqrt{345}+35}{16}$$

The equation is now solved.

$$t=\frac{35-\sqrt{345}}{16} t=\frac{\sqrt{345}+35}{16}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$-16t^{2}+70t+5=60$$

Subtract 5 from both sides.

$$-16t^{2}+70t=60-5$$

Subtract 5 from 60 to get 55.

$$-16t^{2}+70t=55$$

Divide both sides by -16.

$$\frac{-16t^{2}+70t}{-16}=\frac{55}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$t^{2}+\frac{70}{-16}t=\frac{55}{-16}$$

Reduce the fraction \frac{70}{-16} to lowest terms by extracting and canceling out 2.

$$t^{2}-\frac{35}{8}t=\frac{55}{-16}$$

Divide 55 by -16.

$$t^{2}-\frac{35}{8}t=-\frac{55}{16}$$

Divide -\frac{35}{8}, the coefficient of the x term, by 2 to get -\frac{35}{16}. Then add the square of -\frac{35}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{35}{8}t+\left(-\frac{35}{16}\right)^{2}=-\frac{55}{16}+\left(-\frac{35}{16}\right)^{2}$$

Square -\frac{35}{16} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{35}{8}t+\frac{1225}{256}=-\frac{55}{16}+\frac{1225}{256}$$

Add -\frac{55}{16} to \frac{1225}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{35}{8}t+\frac{1225}{256}=\frac{345}{256}$$

Factor t^{2}-\frac{35}{8}t+\frac{1225}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{35}{16}\right)^{2}=\frac{345}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{35}{16}\right)^{2}}=\sqrt{\frac{345}{256}}$$

Simplify.

$$t-\frac{35}{16}=\frac{\sqrt{345}}{16} t-\frac{35}{16}=-\frac{\sqrt{345}}{16}$$

Add \frac{35}{16} to both sides of the equation.

$$t=\frac{\sqrt{345}+35}{16} t=\frac{35-\sqrt{345}}{16}$$