Solve 0=-16t^2+2448t+0

$$0 = - 16 t ^ { 2 } + 2448 t + 0$$

Solve for t

$$t=153$$

$$t=0$$

Steps Using Factoring

Anything plus zero gives itself.

$$0=-16t^{2}+2448t$$

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

$$-16t^{2}+2448t=0$$

Factor out t.

$$t\left(-16t+2448\right)=0$$

To find equation solutions, solve t=0 and -16t+2448=0.

$$t=0 t=153$$

Steps Using the Quadratic Formula

Anything plus zero gives itself.

$$0=-16t^{2}+2448t$$

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

$$-16t^{2}+2448t=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 2448 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-2448±\sqrt{2448^{2}}}{2\left(-16\right)}$$

Take the square root of 2448^{2}.

$$t=\frac{-2448±2448}{2\left(-16\right)}$$

Multiply 2 times -16.

$$t=\frac{-2448±2448}{-32}$$

Now solve the equation t=\frac{-2448±2448}{-32} when ± is plus. Add -2448 to 2448.

$$t=\frac{0}{-32}$$

Divide 0 by -32.

$$t=0$$

Now solve the equation t=\frac{-2448±2448}{-32} when ± is minus. Subtract 2448 from -2448.

$$t=-\frac{4896}{-32}$$

Divide -4896 by -32.

$$t=153$$

The equation is now solved.

$$t=0 t=153$$

Steps for Completing the Square

Anything plus zero gives itself.

$$0=-16t^{2}+2448t$$

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

$$-16t^{2}+2448t=0$$

Divide both sides by -16.

$$\frac{-16t^{2}+2448t}{-16}=\frac{0}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$t^{2}+\frac{2448}{-16}t=\frac{0}{-16}$$

Divide 2448 by -16.

$$t^{2}-153t=\frac{0}{-16}$$

Divide 0 by -16.

$$t^{2}-153t=0$$

Divide -153, the coefficient of the x term, by 2 to get -\frac{153}{2}. Then add the square of -\frac{153}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-153t+\left(-\frac{153}{2}\right)^{2}=\left(-\frac{153}{2}\right)^{2}$$

Square -\frac{153}{2} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-153t+\frac{23409}{4}=\frac{23409}{4}$$

Factor t^{2}-153t+\frac{23409}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{153}{2}\right)^{2}=\frac{23409}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{153}{2}\right)^{2}}=\sqrt{\frac{23409}{4}}$$

Simplify.

$$t-\frac{153}{2}=\frac{153}{2} t-\frac{153}{2}=-\frac{153}{2}$$

Add \frac{153}{2} to both sides of the equation.

$$t=153 t=0$$