Write as a Function of s s=-16t^2+128t

$s = - 16 t^{2} + 128 t$

Step 1

Rewrite the equation as $- 16 t^{2} + 128 t = s$ .

$- 16 t^{2} + 128 t = s$

Step 2

Subtract $s$ from both sides of the equation.

$- 16 t^{2} + 128 t - s = 0$

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 $a = - 16$ , $b = 128$ , and $c = - s$ into the quadratic formula and solve for $t$ .

$\frac{- 128 \pm \sqrt{128^{2} - 4 \cdot (- 16 \cdot (- s))}}{2 \cdot - 16}$

Step 5

Simplify.

Simplify the numerator.

Raise $128$ to the power of $2$ .

$t = \frac{- 128 \pm \sqrt{16384 - 4 \cdot - 16 \cdot (- 1 s)}}{2 \cdot - 16}$

Multiply $- 4 \cdot - 16 \cdot - 1$ .

Multiply $- 4$ by $- 16$ .

$t = \frac{- 128 \pm \sqrt{16384 + 64 \cdot (- 1 s)}}{2 \cdot - 16}$

Multiply $64$ by $- 1$ .

$t = \frac{- 128 \pm \sqrt{16384 - 64 s}}{2 \cdot - 16}$

Factor $64$ out of $16384 - 64 s$ .

Factor $64$ out of $16384$ .

$t = \frac{- 128 \pm \sqrt{64 (256) - 64 s}}{2 \cdot - 16}$

Factor $64$ out of $- 64 s$ .

$t = \frac{- 128 \pm \sqrt{64 (256) + 64 (- s)}}{2 \cdot - 16}$

Factor $64$ out of $64 (256) + 64 (- s)$ .

$t = \frac{- 128 \pm \sqrt{64 (256 - s)}}{2 \cdot - 16}$

Rewrite $64 (256 - s)$ as $8^{2} (16^{2} - s)$ .

Rewrite $64$ as $8^{2}$ .

$t = \frac{- 128 \pm \sqrt{8^{2} (256 - s)}}{2 \cdot - 16}$

Rewrite $256$ as $16^{2}$ .

$t = \frac{- 128 \pm \sqrt{8^{2} (16^{2} - s)}}{2 \cdot - 16}$

Pull terms out from under the radical.

$t = \frac{- 128 \pm 8 \sqrt{16^{2} - s}}{2 \cdot - 16}$

Raise $16$ to the power of $2$ .

$t = \frac{- 128 \pm 8 \sqrt{256 - s}}{2 \cdot - 16}$

Multiply $2$ by $- 16$ .

$t = \frac{- 128 \pm 8 \sqrt{256 - s}}{- 32}$

Simplify $\frac{- 128 \pm 8 \sqrt{256 - s}}{- 32}$ .

$t = \frac{16 \pm \sqrt{256 - s}}{4}$

Step 6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $128$ to the power of $2$ .

$t = \frac{- 128 \pm \sqrt{16384 - 4 \cdot - 16 \cdot (- 1 s)}}{2 \cdot - 16}$

Multiply $- 4 \cdot - 16 \cdot - 1$ .

Multiply $- 4$ by $- 16$ .

$t = \frac{- 128 \pm \sqrt{16384 + 64 \cdot (- 1 s)}}{2 \cdot - 16}$

Multiply $64$ by $- 1$ .

$t = \frac{- 128 \pm \sqrt{16384 - 64 s}}{2 \cdot - 16}$

Factor $64$ out of $16384 - 64 s$ .

Factor $64$ out of $16384$ .

$t = \frac{- 128 \pm \sqrt{64 (256) - 64 s}}{2 \cdot - 16}$

Factor $64$ out of $- 64 s$ .

$t = \frac{- 128 \pm \sqrt{64 (256) + 64 (- s)}}{2 \cdot - 16}$

Factor $64$ out of $64 (256) + 64 (- s)$ .

$t = \frac{- 128 \pm \sqrt{64 (256 - s)}}{2 \cdot - 16}$

Rewrite $64 (256 - s)$ as $8^{2} (16^{2} - s)$ .

Rewrite $64$ as $8^{2}$ .

$t = \frac{- 128 \pm \sqrt{8^{2} (256 - s)}}{2 \cdot - 16}$

Rewrite $256$ as $16^{2}$ .

$t = \frac{- 128 \pm \sqrt{8^{2} (16^{2} - s)}}{2 \cdot - 16}$

Pull terms out from under the radical.

$t = \frac{- 128 \pm 8 \sqrt{16^{2} - s}}{2 \cdot - 16}$

Raise $16$ to the power of $2$ .

$t = \frac{- 128 \pm 8 \sqrt{256 - s}}{2 \cdot - 16}$

Multiply $2$ by $- 16$ .

$t = \frac{- 128 \pm 8 \sqrt{256 - s}}{- 32}$

Simplify $\frac{- 128 \pm 8 \sqrt{256 - s}}{- 32}$ .

$t = \frac{16 \pm \sqrt{256 - s}}{4}$

Change the $\pm$ to $+$ .

$t = \frac{16 + \sqrt{256 - s}}{4}$

Step 7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $128$ to the power of $2$ .

$t = \frac{- 128 \pm \sqrt{16384 - 4 \cdot - 16 \cdot (- 1 s)}}{2 \cdot - 16}$

Multiply $- 4 \cdot - 16 \cdot - 1$ .

Multiply $- 4$ by $- 16$ .

$t = \frac{- 128 \pm \sqrt{16384 + 64 \cdot (- 1 s)}}{2 \cdot - 16}$

Multiply $64$ by $- 1$ .

$t = \frac{- 128 \pm \sqrt{16384 - 64 s}}{2 \cdot - 16}$

Factor $64$ out of $16384 - 64 s$ .

Factor $64$ out of $16384$ .

$t = \frac{- 128 \pm \sqrt{64 (256) - 64 s}}{2 \cdot - 16}$

Factor $64$ out of $- 64 s$ .

$t = \frac{- 128 \pm \sqrt{64 (256) + 64 (- s)}}{2 \cdot - 16}$

Factor $64$ out of $64 (256) + 64 (- s)$ .

$t = \frac{- 128 \pm \sqrt{64 (256 - s)}}{2 \cdot - 16}$

Rewrite $64 (256 - s)$ as $8^{2} (16^{2} - s)$ .

Rewrite $64$ as $8^{2}$ .

$t = \frac{- 128 \pm \sqrt{8^{2} (256 - s)}}{2 \cdot - 16}$

Rewrite $256$ as $16^{2}$ .

$t = \frac{- 128 \pm \sqrt{8^{2} (16^{2} - s)}}{2 \cdot - 16}$

Pull terms out from under the radical.

$t = \frac{- 128 \pm 8 \sqrt{16^{2} - s}}{2 \cdot - 16}$

Raise $16$ to the power of $2$ .

$t = \frac{- 128 \pm 8 \sqrt{256 - s}}{2 \cdot - 16}$

Multiply $2$ by $- 16$ .

$t = \frac{- 128 \pm 8 \sqrt{256 - s}}{- 32}$

Simplify $\frac{- 128 \pm 8 \sqrt{256 - s}}{- 32}$ .

$t = \frac{16 \pm \sqrt{256 - s}}{4}$

Change the $\pm$ to $-$ .

$t = \frac{16 - \sqrt{256 - s}}{4}$

Step 8

The final answer is the combination of both solutions.

$t = \frac{16 + \sqrt{256 - s}}{4}$

$t = \frac{16 - \sqrt{256 - s}}{4}$

Write as a Function of s s=-16t^2+128t

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