Write as a Function of p p=-1/2s^2+280s-1200

$p = - \frac{1}{2} s^{2} + 280 s - 1200$

Step 1

Rewrite the equation as $- \frac{1}{2} s^{2} + 280 s - 1200 = p$ .

$- \frac{1}{2} s^{2} + 280 s - 1200 = p$

Step 2

Combine $s^{2}$ and $\frac{1}{2}$ .

$- \frac{s^{2}}{2} + 280 s - 1200 = p$

Step 3

Subtract $p$ from both sides of the equation.

$- \frac{s^{2}}{2} + 280 s - 1200 - p = 0$

Step 4

Multiply through by the least common denominator $2$ , then simplify.

Apply the distributive property.

$2 (- \frac{s^{2}}{2}) + 2 (280 s) + 2 \cdot - 1200 + 2 (- p) = 0$

Simplify.

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{s^{2}}{2}$ into the numerator.

$2 (\frac{- s^{2}}{2}) + 2 (280 s) + 2 \cdot - 1200 + 2 (- p) = 0$

Cancel the common factor.

$2 (\frac{- s^{2}}{2}) + 2 (280 s) + 2 \cdot - 1200 + 2 (- p) = 0$

Rewrite the expression.

$- s^{2} + 2 (280 s) + 2 \cdot - 1200 + 2 (- p) = 0$

Multiply $280$ by $2$ .

$- s^{2} + 560 s + 2 \cdot - 1200 + 2 (- p) = 0$

Multiply $2$ by $- 1200$ .

$- s^{2} + 560 s - 2400 + 2 (- p) = 0$

Multiply $- 1$ by $2$ .

$- s^{2} + 560 s - 2400 - 2 p = 0$

Move $- 2400$ .

$- s^{2} + 560 s - 2 p - 2400 = 0$

Move $560 s$ .

$- s^{2} - 2 p + 560 s - 2400 = 0$

Step 5

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6

Substitute the values $a = - 1$ , $b = 560$ , and $c = - 2 p - 2400$ into the quadratic formula and solve for $s$ .

$\frac{- 560 \pm \sqrt{560^{2} - 4 \cdot (- 1 \cdot (- 2 p - 2400))}}{2 \cdot - 1}$

Step 7

Simplify.

Simplify the numerator.

Raise $560$ to the power of $2$ .

$s = \frac{- 560 \pm \sqrt{313600 - 4 \cdot - 1 \cdot (- 2 p - 2400)}}{2 \cdot - 1}$

Multiply $- 4$ by $- 1$ .

$s = \frac{- 560 \pm \sqrt{313600 + 4 \cdot (- 2 p - 2400)}}{2 \cdot - 1}$

Apply the distributive property.

$s = \frac{- 560 \pm \sqrt{313600 + 4 (- 2 p) + 4 \cdot - 2400}}{2 \cdot - 1}$

Multiply $- 2$ by $4$ .

$s = \frac{- 560 \pm \sqrt{313600 - 8 p + 4 \cdot - 2400}}{2 \cdot - 1}$

Multiply $4$ by $- 2400$ .

$s = \frac{- 560 \pm \sqrt{313600 - 8 p - 9600}}{2 \cdot - 1}$

Subtract $9600$ from $313600$ .

$s = \frac{- 560 \pm \sqrt{- 8 p + 304000}}{2 \cdot - 1}$

Factor $8$ out of $- 8 p + 304000$ .

Factor $8$ out of $- 8 p$ .

$s = \frac{- 560 \pm \sqrt{8 (- p) + 304000}}{2 \cdot - 1}$

Factor $8$ out of $304000$ .

$s = \frac{- 560 \pm \sqrt{8 (- p) + 8 (38000)}}{2 \cdot - 1}$

Factor $8$ out of $8 (- p) + 8 (38000)$ .

$s = \frac{- 560 \pm \sqrt{8 (- p + 38000)}}{2 \cdot - 1}$

Rewrite $8 (- p + 38000)$ as $2^{2} \cdot (2 (- p + 38000))$ .

Factor $4$ out of $8$ .

$s = \frac{- 560 \pm \sqrt{4 (2) (- p + 38000)}}{2 \cdot - 1}$

Rewrite $4$ as $2^{2}$ .

$s = \frac{- 560 \pm \sqrt{2^{2} \cdot (2 (- p + 38000))}}{2 \cdot - 1}$

Add parentheses.

$s = \frac{- 560 \pm \sqrt{2^{2} \cdot (2 (- p + 38000))}}{2 \cdot - 1}$

Pull terms out from under the radical.

$s = \frac{- 560 \pm 2 \sqrt{2 (- p + 38000)}}{2 \cdot - 1}$

Multiply $2$ by $- 1$ .

$s = \frac{- 560 \pm 2 \sqrt{2 (- p + 38000)}}{- 2}$

Simplify $\frac{- 560 \pm 2 \sqrt{2 (- p + 38000)}}{- 2}$ .

$s = 280 \pm \sqrt{2 (- p + 38000)}$

Step 8

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

Simplify the numerator.

Raise $560$ to the power of $2$ .

$s = \frac{- 560 \pm \sqrt{313600 - 4 \cdot - 1 \cdot (- 2 p - 2400)}}{2 \cdot - 1}$

Multiply $- 4$ by $- 1$ .

$s = \frac{- 560 \pm \sqrt{313600 + 4 \cdot (- 2 p - 2400)}}{2 \cdot - 1}$

Apply the distributive property.

$s = \frac{- 560 \pm \sqrt{313600 + 4 (- 2 p) + 4 \cdot - 2400}}{2 \cdot - 1}$

Multiply $- 2$ by $4$ .

$s = \frac{- 560 \pm \sqrt{313600 - 8 p + 4 \cdot - 2400}}{2 \cdot - 1}$

Multiply $4$ by $- 2400$ .

$s = \frac{- 560 \pm \sqrt{313600 - 8 p - 9600}}{2 \cdot - 1}$

Subtract $9600$ from $313600$ .

$s = \frac{- 560 \pm \sqrt{- 8 p + 304000}}{2 \cdot - 1}$

Factor $8$ out of $- 8 p + 304000$ .

Factor $8$ out of $- 8 p$ .

$s = \frac{- 560 \pm \sqrt{8 (- p) + 304000}}{2 \cdot - 1}$

Factor $8$ out of $304000$ .

$s = \frac{- 560 \pm \sqrt{8 (- p) + 8 (38000)}}{2 \cdot - 1}$

Factor $8$ out of $8 (- p) + 8 (38000)$ .

$s = \frac{- 560 \pm \sqrt{8 (- p + 38000)}}{2 \cdot - 1}$

Rewrite $8 (- p + 38000)$ as $2^{2} \cdot (2 (- p + 38000))$ .

Factor $4$ out of $8$ .

$s = \frac{- 560 \pm \sqrt{4 (2) (- p + 38000)}}{2 \cdot - 1}$

Rewrite $4$ as $2^{2}$ .

$s = \frac{- 560 \pm \sqrt{2^{2} \cdot (2 (- p + 38000))}}{2 \cdot - 1}$

Add parentheses.

$s = \frac{- 560 \pm \sqrt{2^{2} \cdot (2 (- p + 38000))}}{2 \cdot - 1}$

Pull terms out from under the radical.

$s = \frac{- 560 \pm 2 \sqrt{2 (- p + 38000)}}{2 \cdot - 1}$

Multiply $2$ by $- 1$ .

$s = \frac{- 560 \pm 2 \sqrt{2 (- p + 38000)}}{- 2}$

Simplify $\frac{- 560 \pm 2 \sqrt{2 (- p + 38000)}}{- 2}$ .

$s = 280 \pm \sqrt{2 (- p + 38000)}$

Change the $\pm$ to $+$ .

$s = 280 + \sqrt{2 (- p + 38000)}$

Step 9

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

Simplify the numerator.

Raise $560$ to the power of $2$ .

$s = \frac{- 560 \pm \sqrt{313600 - 4 \cdot - 1 \cdot (- 2 p - 2400)}}{2 \cdot - 1}$

Multiply $- 4$ by $- 1$ .

$s = \frac{- 560 \pm \sqrt{313600 + 4 \cdot (- 2 p - 2400)}}{2 \cdot - 1}$

Apply the distributive property.

$s = \frac{- 560 \pm \sqrt{313600 + 4 (- 2 p) + 4 \cdot - 2400}}{2 \cdot - 1}$

Multiply $- 2$ by $4$ .

$s = \frac{- 560 \pm \sqrt{313600 - 8 p + 4 \cdot - 2400}}{2 \cdot - 1}$

Multiply $4$ by $- 2400$ .

$s = \frac{- 560 \pm \sqrt{313600 - 8 p - 9600}}{2 \cdot - 1}$

Subtract $9600$ from $313600$ .

$s = \frac{- 560 \pm \sqrt{- 8 p + 304000}}{2 \cdot - 1}$

Factor $8$ out of $- 8 p + 304000$ .

Factor $8$ out of $- 8 p$ .

$s = \frac{- 560 \pm \sqrt{8 (- p) + 304000}}{2 \cdot - 1}$

Factor $8$ out of $304000$ .

$s = \frac{- 560 \pm \sqrt{8 (- p) + 8 (38000)}}{2 \cdot - 1}$

Factor $8$ out of $8 (- p) + 8 (38000)$ .

$s = \frac{- 560 \pm \sqrt{8 (- p + 38000)}}{2 \cdot - 1}$

Rewrite $8 (- p + 38000)$ as $2^{2} \cdot (2 (- p + 38000))$ .

Factor $4$ out of $8$ .

$s = \frac{- 560 \pm \sqrt{4 (2) (- p + 38000)}}{2 \cdot - 1}$

Rewrite $4$ as $2^{2}$ .

$s = \frac{- 560 \pm \sqrt{2^{2} \cdot (2 (- p + 38000))}}{2 \cdot - 1}$

Add parentheses.

$s = \frac{- 560 \pm \sqrt{2^{2} \cdot (2 (- p + 38000))}}{2 \cdot - 1}$

Pull terms out from under the radical.

$s = \frac{- 560 \pm 2 \sqrt{2 (- p + 38000)}}{2 \cdot - 1}$

Multiply $2$ by $- 1$ .

$s = \frac{- 560 \pm 2 \sqrt{2 (- p + 38000)}}{- 2}$

Simplify $\frac{- 560 \pm 2 \sqrt{2 (- p + 38000)}}{- 2}$ .

$s = 280 \pm \sqrt{2 (- p + 38000)}$

Change the $\pm$ to $-$ .

$s = 280 - \sqrt{2 (- p + 38000)}$

Step 10

The final answer is the combination of both solutions.

$s = 280 + \sqrt{2 (- p + 38000)}$

$s = 280 - \sqrt{2 (- p + 38000)}$

Write as a Function of p p=-1/2s^2+280s-1200

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