Algebra Examples

$p = - 25 x^{2} + 300 x$

Step 1

Rewrite the equation as $- 25 x^{2} + 300 x = p$ .

$- 25 x^{2} + 300 x = p$

Step 2

Subtract $p$ from both sides of the equation.

$- 25 x^{2} + 300 x - p = 0$

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 $a = - 25$ , $b = 300$ , and $c = - p$ into the quadratic formula and solve for $x$ .

$\frac{- 300 \pm \sqrt{300^{2} - 4 \cdot (- 25 \cdot (- p))}}{2 \cdot - 25}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $300$ to the power of $2$ .

$x = \frac{- 300 \pm \sqrt{90000 - 4 \cdot - 25 \cdot (- 1 p)}}{2 \cdot - 25}$

Step 5.1.2

Multiply $- 4 \cdot - 25 \cdot - 1$ .

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Step 5.1.2.1

Multiply $- 4$ by $- 25$ .

$x = \frac{- 300 \pm \sqrt{90000 + 100 \cdot (- 1 p)}}{2 \cdot - 25}$

Step 5.1.2.2

Multiply $100$ by $- 1$ .

$x = \frac{- 300 \pm \sqrt{90000 - 100 p}}{2 \cdot - 25}$

Step 5.1.3

Factor $100$ out of $90000 - 100 p$ .

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Step 5.1.3.1

Factor $100$ out of $90000$ .

$x = \frac{- 300 \pm \sqrt{100 (900) - 100 p}}{2 \cdot - 25}$

Step 5.1.3.2

Factor $100$ out of $- 100 p$ .

$x = \frac{- 300 \pm \sqrt{100 (900) + 100 (- p)}}{2 \cdot - 25}$

Step 5.1.3.3

Factor $100$ out of $100 (900) + 100 (- p)$ .

$x = \frac{- 300 \pm \sqrt{100 (900 - p)}}{2 \cdot - 25}$

Step 5.1.4

Rewrite $100 (900 - p)$ as $10^{2} (30^{2} - p)$ .

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Step 5.1.4.1

Rewrite $100$ as $10^{2}$ .

$x = \frac{- 300 \pm \sqrt{10^{2} (900 - p)}}{2 \cdot - 25}$

Step 5.1.4.2

Rewrite $900$ as $30^{2}$ .

$x = \frac{- 300 \pm \sqrt{10^{2} (30^{2} - p)}}{2 \cdot - 25}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{- 300 \pm 10 \sqrt{30^{2} - p}}{2 \cdot - 25}$

Step 5.1.6

Raise $30$ to the power of $2$ .

$x = \frac{- 300 \pm 10 \sqrt{900 - p}}{2 \cdot - 25}$

Step 5.2

Multiply $2$ by $- 25$ .

$x = \frac{- 300 \pm 10 \sqrt{900 - p}}{- 50}$

Step 5.3

Simplify $\frac{- 300 \pm 10 \sqrt{900 - p}}{- 50}$ .

$x = \frac{30 \pm \sqrt{900 - p}}{5}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{30 + \sqrt{900 - p}}{5}$

$x = \frac{30 - \sqrt{900 - p}}{5}$