Solve 120=-16t^2+95t

$$120 = - 16 t ^ { 2 } + 95 t$$

Solve for t

$$t = \frac{\sqrt{1345} + 95}{32} \approx 4.114820051$$

$$t = \frac{95 - \sqrt{1345}}{32} \approx 1.822679949$$

Steps Using the Quadratic Formula

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

$$-16t^{2}+95t=120$$

Subtract 120 from both sides.

$$-16t^{2}+95t-120=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 95 for b, and -120 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-95±\sqrt{95^{2}-4\left(-16\right)\left(-120\right)}}{2\left(-16\right)}$$

Square 95.

$$t=\frac{-95±\sqrt{9025-4\left(-16\right)\left(-120\right)}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$t=\frac{-95±\sqrt{9025+64\left(-120\right)}}{2\left(-16\right)}$$

Multiply 64 times -120.

$$t=\frac{-95±\sqrt{9025-7680}}{2\left(-16\right)}$$

Add 9025 to -7680.

$$t=\frac{-95±\sqrt{1345}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$t=\frac{-95±\sqrt{1345}}{-32}$$

Now solve the equation t=\frac{-95±\sqrt{1345}}{-32} when ± is plus. Add -95 to \sqrt{1345}.

$$t=\frac{\sqrt{1345}-95}{-32}$$

Divide -95+\sqrt{1345} by -32.

$$t=\frac{95-\sqrt{1345}}{32}$$

Now solve the equation t=\frac{-95±\sqrt{1345}}{-32} when ± is minus. Subtract \sqrt{1345} from -95.

$$t=\frac{-\sqrt{1345}-95}{-32}$$

Divide -95-\sqrt{1345} by -32.

$$t=\frac{\sqrt{1345}+95}{32}$$

The equation is now solved.

$$t=\frac{95-\sqrt{1345}}{32} t=\frac{\sqrt{1345}+95}{32}$$

Steps for Completing the Square

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

$$-16t^{2}+95t=120$$

Divide both sides by -16.

$$\frac{-16t^{2}+95t}{-16}=\frac{120}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$t^{2}+\frac{95}{-16}t=\frac{120}{-16}$$

Divide 95 by -16.

$$t^{2}-\frac{95}{16}t=\frac{120}{-16}$$

Reduce the fraction \frac{120}{-16} to lowest terms by extracting and canceling out 8.

$$t^{2}-\frac{95}{16}t=-\frac{15}{2}$$

Divide -\frac{95}{16}, the coefficient of the x term, by 2 to get -\frac{95}{32}. Then add the square of -\frac{95}{32} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{95}{16}t+\left(-\frac{95}{32}\right)^{2}=-\frac{15}{2}+\left(-\frac{95}{32}\right)^{2}$$

Square -\frac{95}{32} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{95}{16}t+\frac{9025}{1024}=-\frac{15}{2}+\frac{9025}{1024}$$

Add -\frac{15}{2} to \frac{9025}{1024} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{95}{16}t+\frac{9025}{1024}=\frac{1345}{1024}$$

Factor t^{2}-\frac{95}{16}t+\frac{9025}{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}.

$$\left(t-\frac{95}{32}\right)^{2}=\frac{1345}{1024}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{95}{32}\right)^{2}}=\sqrt{\frac{1345}{1024}}$$

Simplify.

$$t-\frac{95}{32}=\frac{\sqrt{1345}}{32} t-\frac{95}{32}=-\frac{\sqrt{1345}}{32}$$

Add \frac{95}{32} to both sides of the equation.

$$t=\frac{\sqrt{1345}+95}{32} t=\frac{95-\sqrt{1345}}{32}$$