Solve 0=-16t^2+12t+1900

$$0 = - 16 t ^ { 2 } + 12 t + 1900$$

Solve for t

$$t = \frac{\sqrt{7609} + 3}{8} \approx 11.278697767$$

$$t=\frac{3-\sqrt{7609}}{8}\approx -10.528697767$$

Steps Using the Quadratic Formula

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

$$-16t^{2}+12t+1900=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 12 for b, and 1900 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-12±\sqrt{12^{2}-4\left(-16\right)\times 1900}}{2\left(-16\right)}$$

Square 12.

$$t=\frac{-12±\sqrt{144-4\left(-16\right)\times 1900}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$t=\frac{-12±\sqrt{144+64\times 1900}}{2\left(-16\right)}$$

Multiply 64 times 1900.

$$t=\frac{-12±\sqrt{144+121600}}{2\left(-16\right)}$$

Add 144 to 121600.

$$t=\frac{-12±\sqrt{121744}}{2\left(-16\right)}$$

Take the square root of 121744.

$$t=\frac{-12±4\sqrt{7609}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$t=\frac{-12±4\sqrt{7609}}{-32}$$

Now solve the equation t=\frac{-12±4\sqrt{7609}}{-32} when ± is plus. Add -12 to 4\sqrt{7609}.

$$t=\frac{4\sqrt{7609}-12}{-32}$$

Divide -12+4\sqrt{7609} by -32.

$$t=\frac{3-\sqrt{7609}}{8}$$

Now solve the equation t=\frac{-12±4\sqrt{7609}}{-32} when ± is minus. Subtract 4\sqrt{7609} from -12.

$$t=\frac{-4\sqrt{7609}-12}{-32}$$

Divide -12-4\sqrt{7609} by -32.

$$t=\frac{\sqrt{7609}+3}{8}$$

The equation is now solved.

$$t=\frac{3-\sqrt{7609}}{8} t=\frac{\sqrt{7609}+3}{8}$$

Steps for Completing the Square

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

$$-16t^{2}+12t+1900=0$$

Subtract 1900 from both sides. Anything subtracted from zero gives its negation.

$$-16t^{2}+12t=-1900$$

Divide both sides by -16.

$$\frac{-16t^{2}+12t}{-16}=-\frac{1900}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$t^{2}+\frac{12}{-16}t=-\frac{1900}{-16}$$

Reduce the fraction \frac{12}{-16} to lowest terms by extracting and canceling out 4.

$$t^{2}-\frac{3}{4}t=-\frac{1900}{-16}$$

Reduce the fraction \frac{-1900}{-16} to lowest terms by extracting and canceling out 4.

$$t^{2}-\frac{3}{4}t=\frac{475}{4}$$

Divide -\frac{3}{4}, the coefficient of the x term, by 2 to get -\frac{3}{8}. Then add the square of -\frac{3}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{3}{4}t+\left(-\frac{3}{8}\right)^{2}=\frac{475}{4}+\left(-\frac{3}{8}\right)^{2}$$

Square -\frac{3}{8} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{3}{4}t+\frac{9}{64}=\frac{475}{4}+\frac{9}{64}$$

Add \frac{475}{4} to \frac{9}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{3}{4}t+\frac{9}{64}=\frac{7609}{64}$$

Factor t^{2}-\frac{3}{4}t+\frac{9}{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}.

$$\left(t-\frac{3}{8}\right)^{2}=\frac{7609}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{3}{8}\right)^{2}}=\sqrt{\frac{7609}{64}}$$

Simplify.

$$t-\frac{3}{8}=\frac{\sqrt{7609}}{8} t-\frac{3}{8}=-\frac{\sqrt{7609}}{8}$$

Add \frac{3}{8} to both sides of the equation.

$$t=\frac{\sqrt{7609}+3}{8} t=\frac{3-\sqrt{7609}}{8}$$