Solve 0=-16t^2+20t+900

$$0 = - 16 t ^ { 2 } + 20 t + 900$$

Solve for t

$$t = \frac{5 \sqrt{145} + 5}{8} \approx 8.150996612$$

$$t=\frac{5-5\sqrt{145}}{8}\approx -6.900996612$$

Steps Using the Quadratic Formula

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

$$-16t^{2}+20t+900=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 20 for b, and 900 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-20±\sqrt{20^{2}-4\left(-16\right)\times 900}}{2\left(-16\right)}$$

Square 20.

$$t=\frac{-20±\sqrt{400-4\left(-16\right)\times 900}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$t=\frac{-20±\sqrt{400+64\times 900}}{2\left(-16\right)}$$

Multiply 64 times 900.

$$t=\frac{-20±\sqrt{400+57600}}{2\left(-16\right)}$$

Add 400 to 57600.

$$t=\frac{-20±\sqrt{58000}}{2\left(-16\right)}$$

Take the square root of 58000.

$$t=\frac{-20±20\sqrt{145}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$t=\frac{-20±20\sqrt{145}}{-32}$$

Now solve the equation t=\frac{-20±20\sqrt{145}}{-32} when ± is plus. Add -20 to 20\sqrt{145}.

$$t=\frac{20\sqrt{145}-20}{-32}$$

Divide -20+20\sqrt{145} by -32.

$$t=\frac{5-5\sqrt{145}}{8}$$

Now solve the equation t=\frac{-20±20\sqrt{145}}{-32} when ± is minus. Subtract 20\sqrt{145} from -20.

$$t=\frac{-20\sqrt{145}-20}{-32}$$

Divide -20-20\sqrt{145} by -32.

$$t=\frac{5\sqrt{145}+5}{8}$$

The equation is now solved.

$$t=\frac{5-5\sqrt{145}}{8} t=\frac{5\sqrt{145}+5}{8}$$

Steps for Completing the Square

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

$$-16t^{2}+20t+900=0$$

Subtract 900 from both sides. Anything subtracted from zero gives its negation.

$$-16t^{2}+20t=-900$$

Divide both sides by -16.

$$\frac{-16t^{2}+20t}{-16}=-\frac{900}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$t^{2}+\frac{20}{-16}t=-\frac{900}{-16}$$

Reduce the fraction \frac{20}{-16} to lowest terms by extracting and canceling out 4.

$$t^{2}-\frac{5}{4}t=-\frac{900}{-16}$$

Reduce the fraction \frac{-900}{-16} to lowest terms by extracting and canceling out 4.

$$t^{2}-\frac{5}{4}t=\frac{225}{4}$$

Divide -\frac{5}{4}, the coefficient of the x term, by 2 to get -\frac{5}{8}. Then add the square of -\frac{5}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{5}{4}t+\left(-\frac{5}{8}\right)^{2}=\frac{225}{4}+\left(-\frac{5}{8}\right)^{2}$$

Square -\frac{5}{8} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{5}{4}t+\frac{25}{64}=\frac{225}{4}+\frac{25}{64}$$

Add \frac{225}{4} to \frac{25}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{5}{4}t+\frac{25}{64}=\frac{3625}{64}$$

Factor t^{2}-\frac{5}{4}t+\frac{25}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{5}{8}\right)^{2}=\frac{3625}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{5}{8}\right)^{2}}=\sqrt{\frac{3625}{64}}$$

Simplify.

$$t-\frac{5}{8}=\frac{5\sqrt{145}}{8} t-\frac{5}{8}=-\frac{5\sqrt{145}}{8}$$

Add \frac{5}{8} to both sides of the equation.

$$t=\frac{5\sqrt{145}+5}{8} t=\frac{5-5\sqrt{145}}{8}$$