Solve for t 97=215t-16t^2

$97 = 215 t - 16 t^{2}$

Step 1

Rewrite the equation as $215 t - 16 t^{2} = 97$ .

$215 t - 16 t^{2} = 97$

Step 2

Subtract $97$ from both sides of the equation.

$215 t - 16 t^{2} - 97 = 0$

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 $a = - 16$ , $b = 215$ , and $c = - 97$ into the quadratic formula and solve for $t$ .

$\frac{- 215 \pm \sqrt{215^{2} - 4 \cdot (- 16 \cdot - 97)}}{2 \cdot - 16}$

Step 5

Simplify.

Simplify the numerator.

Raise $215$ to the power of $2$ .

$t = \frac{- 215 \pm \sqrt{46225 - 4 \cdot - 16 \cdot - 97}}{2 \cdot - 16}$

Multiply $- 4 \cdot - 16 \cdot - 97$ .

Multiply $- 4$ by $- 16$ .

$t = \frac{- 215 \pm \sqrt{46225 + 64 \cdot - 97}}{2 \cdot - 16}$

Multiply $64$ by $- 97$ .

$t = \frac{- 215 \pm \sqrt{46225 - 6208}}{2 \cdot - 16}$

Subtract $6208$ from $46225$ .

$t = \frac{- 215 \pm \sqrt{40017}}{2 \cdot - 16}$

Multiply $2$ by $- 16$ .

$t = \frac{- 215 \pm \sqrt{40017}}{- 32}$

Simplify $\frac{- 215 \pm \sqrt{40017}}{- 32}$ .

$t = \frac{215 \pm \sqrt{40017}}{32}$

Step 6

The final answer is the combination of both solutions.

$t = \frac{215 + \sqrt{40017}}{32}, \frac{215 - \sqrt{40017}}{32}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$t = \frac{215 + \sqrt{40017}}{32}, \frac{215 - \sqrt{40017}}{32}$

Decimal Form:

$t = 12.97007798 \dots, 0.46742201 \dots$