Solve for t 34=105t-16t^2

$34 = 105 t - 16 t^{2}$

Step 1

Rewrite the equation as $105 t - 16 t^{2} = 34$ .

$105 t - 16 t^{2} = 34$

Step 2

Subtract $34$ from both sides of the equation.

$105 t - 16 t^{2} - 34 = 0$

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 $a = - 16$ , $b = 105$ , and $c = - 34$ into the quadratic formula and solve for $t$ .

$\frac{- 105 \pm \sqrt{105^{2} - 4 \cdot (- 16 \cdot - 34)}}{2 \cdot - 16}$

Step 5

Simplify.

Simplify the numerator.

Raise $105$ to the power of $2$ .

$t = \frac{- 105 \pm \sqrt{11025 - 4 \cdot - 16 \cdot - 34}}{2 \cdot - 16}$

Multiply $- 4 \cdot - 16 \cdot - 34$ .

Multiply $- 4$ by $- 16$ .

$t = \frac{- 105 \pm \sqrt{11025 + 64 \cdot - 34}}{2 \cdot - 16}$

Multiply $64$ by $- 34$ .

$t = \frac{- 105 \pm \sqrt{11025 - 2176}}{2 \cdot - 16}$

Subtract $2176$ from $11025$ .

$t = \frac{- 105 \pm \sqrt{8849}}{2 \cdot - 16}$

Multiply $2$ by $- 16$ .

$t = \frac{- 105 \pm \sqrt{8849}}{- 32}$

Simplify $\frac{- 105 \pm \sqrt{8849}}{- 32}$ .

$t = \frac{105 \pm \sqrt{8849}}{32}$

Step 6

The final answer is the combination of both solutions.

$t = \frac{105 + \sqrt{8849}}{32}, \frac{105 - \sqrt{8849}}{32}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$t = \frac{105 + \sqrt{8849}}{32}, \frac{105 - \sqrt{8849}}{32}$

Decimal Form:

$t = 6.22091011 \dots, 0.34158988 \dots$