Solve for t h=255-21t-16t^2

$h = 255 - 21 t - 16 t^{2}$

Step 1

Rewrite the equation as $255 - 21 t - 16 t^{2} = h$ .

$255 - 21 t - 16 t^{2} = h$

Step 2

Subtract $h$ from both sides of the equation.

$255 - 21 t - 16 t^{2} - h = 0$

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 $a = - 16$ , $b = - 21$ , and $c = 255 - h$ into the quadratic formula and solve for $t$ .

$\frac{21 \pm \sqrt{{(- 21)}^{2} - 4 \cdot (- 16 \cdot (255 - h))}}{2 \cdot - 16}$

Step 5

Simplify.

Simplify the numerator.

Raise $- 21$ to the power of $2$ .

$t = \frac{21 \pm \sqrt{441 - 4 \cdot - 16 \cdot (255 - h)}}{2 \cdot - 16}$

Multiply $- 4$ by $- 16$ .

$t = \frac{21 \pm \sqrt{441 + 64 \cdot (255 - h)}}{2 \cdot - 16}$

Apply the distributive property.

$t = \frac{21 \pm \sqrt{441 + 64 \cdot 255 + 64 (- h)}}{2 \cdot - 16}$

Multiply $64$ by $255$ .

$t = \frac{21 \pm \sqrt{441 + 16320 + 64 (- h)}}{2 \cdot - 16}$

Multiply $- 1$ by $64$ .

$t = \frac{21 \pm \sqrt{441 + 16320 - 64 h}}{2 \cdot - 16}$

Add $441$ and $16320$ .

$t = \frac{21 \pm \sqrt{16761 - 64 h}}{2 \cdot - 16}$

Multiply $2$ by $- 16$ .

$t = \frac{21 \pm \sqrt{16761 - 64 h}}{- 32}$

Move the negative in front of the fraction.

$t = - \frac{21 \pm \sqrt{16761 - 64 h}}{32}$

Step 6

The final answer is the combination of both solutions.

$t = - \frac{21 + \sqrt{16761 - 64 h}}{32}$

$t = - \frac{21 - \sqrt{16761 - 64 h}}{32}$