Solve Using the Quadratic Formula 160=-16t^2+112t

$160 = - 16 t^{2} + 112 t$

Step 1

Move all the expressions to the left side of the equation.

Add $16 t^{2}$ to both sides of the equation.

$160 + 16 t^{2} = 112 t$

Subtract $112 t$ from both sides of the equation.

$160 + 16 t^{2} - 112 t = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 16$ , $b = - 112$ , and $c = 160$ into the quadratic formula and solve for $t$ .

$\frac{112 \pm \sqrt{{(- 112)}^{2} - 4 \cdot (16 \cdot 160)}}{2 \cdot 16}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 112$ to the power of $2$ .

$t = \frac{112 \pm \sqrt{12544 - 4 \cdot 16 \cdot 160}}{2 \cdot 16}$

Multiply $- 4 \cdot 16 \cdot 160$ .

Multiply $- 4$ by $16$ .

$t = \frac{112 \pm \sqrt{12544 - 64 \cdot 160}}{2 \cdot 16}$

Multiply $- 64$ by $160$ .

$t = \frac{112 \pm \sqrt{12544 - 10240}}{2 \cdot 16}$

Subtract $10240$ from $12544$ .

$t = \frac{112 \pm \sqrt{2304}}{2 \cdot 16}$

Rewrite $2304$ as $48^{2}$ .

$t = \frac{112 \pm \sqrt{48^{2}}}{2 \cdot 16}$

Pull terms out from under the radical, assuming positive real numbers.

$t = \frac{112 \pm 48}{2 \cdot 16}$

Multiply $2$ by $16$ .

$t = \frac{112 \pm 48}{32}$

Simplify $\frac{112 \pm 48}{32}$ .

$t = \frac{7 \pm 3}{2}$

Step 5

The final answer is the combination of both solutions.

$t = 5, 2$