Solve for t h=105-9t-16t^2

$h = 105 - 9 t - 16 t^{2}$

Step 1

Rewrite the equation as $105 - 9 t - 16 t^{2} = h$ .

$105 - 9 t - 16 t^{2} = h$

Step 2

Subtract $h$ from both sides of the equation.

$105 - 9 t - 16 t^{2} - h = 0$

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 $a = - 16$ , $b = - 9$ , and $c = 105 - h$ into the quadratic formula and solve for $t$ .

$\frac{9 \pm \sqrt{{(- 9)}^{2} - 4 \cdot (- 16 \cdot (105 - h))}}{2 \cdot - 16}$

Step 5

Simplify.

Simplify the numerator.

Raise $- 9$ to the power of $2$ .

$t = \frac{9 \pm \sqrt{81 - 4 \cdot - 16 \cdot (105 - h)}}{2 \cdot - 16}$

Multiply $- 4$ by $- 16$ .

$t = \frac{9 \pm \sqrt{81 + 64 \cdot (105 - h)}}{2 \cdot - 16}$

Apply the distributive property.

$t = \frac{9 \pm \sqrt{81 + 64 \cdot 105 + 64 (- h)}}{2 \cdot - 16}$

Multiply $64$ by $105$ .

$t = \frac{9 \pm \sqrt{81 + 6720 + 64 (- h)}}{2 \cdot - 16}$

Multiply $- 1$ by $64$ .

$t = \frac{9 \pm \sqrt{81 + 6720 - 64 h}}{2 \cdot - 16}$

Add $81$ and $6720$ .

$t = \frac{9 \pm \sqrt{6801 - 64 h}}{2 \cdot - 16}$

Multiply $2$ by $- 16$ .

$t = \frac{9 \pm \sqrt{6801 - 64 h}}{- 32}$

Move the negative in front of the fraction.

$t = - \frac{9 \pm \sqrt{6801 - 64 h}}{32}$

Step 6

The final answer is the combination of both solutions.

$t = - \frac{9 + \sqrt{6801 - 64 h}}{32}$

$t = - \frac{9 - \sqrt{6801 - 64 h}}{32}$