Solve for t 87=195t-16t^2

$87 = 195 t - 16 t^{2}$

Step 1

Rewrite the equation as $195 t - 16 t^{2} = 87$ .

$195 t - 16 t^{2} = 87$

Step 2

Subtract $87$ from both sides of the equation.

$195 t - 16 t^{2} - 87 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = - 16$ , $b = 195$ , and $c = - 87$ into the quadratic formula and solve for $t$ .

$\frac{- 195 \pm \sqrt{195^{2} - 4 \cdot (- 16 \cdot - 87)}}{2 \cdot - 16}$

Step 5

Simplify.

Simplify the numerator.

Raise $195$ to the power of $2$ .

$t = \frac{- 195 \pm \sqrt{38025 - 4 \cdot - 16 \cdot - 87}}{2 \cdot - 16}$

Multiply $- 4 \cdot - 16 \cdot - 87$ .

Multiply $- 4$ by $- 16$ .

$t = \frac{- 195 \pm \sqrt{38025 + 64 \cdot - 87}}{2 \cdot - 16}$

Multiply $64$ by $- 87$ .

$t = \frac{- 195 \pm \sqrt{38025 - 5568}}{2 \cdot - 16}$

Subtract $5568$ from $38025$ .

$t = \frac{- 195 \pm \sqrt{32457}}{2 \cdot - 16}$

Multiply $2$ by $- 16$ .

$t = \frac{- 195 \pm \sqrt{32457}}{- 32}$

Simplify $\frac{- 195 \pm \sqrt{32457}}{- 32}$ .

$t = \frac{195 \pm \sqrt{32457}}{32}$

Step 6

The final answer is the combination of both solutions.

$t = \frac{195 + \sqrt{32457}}{32}, \frac{195 - \sqrt{32457}}{32}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$t = \frac{195 + \sqrt{32457}}{32}, \frac{195 - \sqrt{32457}}{32}$

Decimal Form:

$t = 11.72369574 \dots, 0.46380425 \dots$