Finite Math Examples

$- 3 = p \cdot 2 + \frac{500}{7 (p \cdot 2 - 500)}$

Step 1

Factor each term.

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Step 1.1

Factor.

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Step 1.1.1

Factor $2$ out of $p \cdot 2 - 500$ .

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Step 1.1.1.1

Factor $2$ out of $p \cdot 2$ .

$- 3 = p \cdot 2 + \frac{500}{7 (2 \cdot p - 500)}$

Step 1.1.1.2

Factor $2$ out of $- 500$ .

$- 3 = p \cdot 2 + \frac{500}{7 (2 \cdot p + 2 \cdot - 250)}$

Step 1.1.1.3

Factor $2$ out of $2 \cdot p + 2 \cdot - 250$ .

$- 3 = p \cdot 2 + \frac{500}{7 (2 (p - 250))}$

Step 1.1.2

Remove unnecessary parentheses.

$- 3 = p \cdot 2 + \frac{500}{7 \cdot 2 (p - 250)}$

Step 1.2

Multiply $7$ by $2$ .

$- 3 = p \cdot 2 + \frac{500}{14 (p - 250)}$

Step 1.3

Reduce the expression $\frac{500}{14 (p - 250)}$ by cancelling the common factors.

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Step 1.3.1

Factor $2$ out of $500$ .

$- 3 = p \cdot 2 + \frac{2 (250)}{14 (p - 250)}$

Step 1.3.2

Factor $2$ out of $14 (p - 250)$ .

$- 3 = p \cdot 2 + \frac{2 (250)}{2 (7 (p - 250))}$

Step 1.3.3

Cancel the common factor.

$- 3 = p \cdot 2 + \frac{2 \cdot 250}{2 (7 (p - 250))}$

Step 1.3.4

Rewrite the expression.

$- 3 = p \cdot 2 + \frac{250}{7 (p - 250)}$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, 1, 1, 7 (p - 250)$

Step 2.2

The LCM of one and any expression is the expression.

$7 (p - 250)$

Step 3

Multiply each term in $- 3 = p \cdot 2 + \frac{250}{7 (p - 250)}$ by $7 (p - 250)$ to eliminate the fractions.

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Step 3.1

Multiply each term in $- 3 = p \cdot 2 + \frac{250}{7 (p - 250)}$ by $7 (p - 250)$ .

$- 3 (7 (p - 250)) = p \cdot 2 (7 (p - 250)) + \frac{250}{7 (p - 250)} (7 (p - 250))$

Step 3.2

Simplify the left side.

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Step 3.2.1

Apply the distributive property.

$- 3 (7 p + 7 \cdot - 250) = p \cdot 2 (7 (p - 250)) + \frac{250}{7 (p - 250)} (7 (p - 250))$

Step 3.2.2

Multiply $7$ by $- 250$ .

$- 3 (7 p - 1750) = p \cdot 2 (7 (p - 250)) + \frac{250}{7 (p - 250)} (7 (p - 250))$

Step 3.2.3

Apply the distributive property.

$- 3 (7 p) - 3 \cdot - 1750 = p \cdot 2 (7 (p - 250)) + \frac{250}{7 (p - 250)} (7 (p - 250))$

Step 3.2.4

Multiply.

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Step 3.2.4.1

Multiply $7$ by $- 3$ .

$- 21 p - 3 \cdot - 1750 = p \cdot 2 (7 (p - 250)) + \frac{250}{7 (p - 250)} (7 (p - 250))$

Step 3.2.4.2

Multiply $- 3$ by $- 1750$ .

$- 21 p + 5250 = p \cdot 2 (7 (p - 250)) + \frac{250}{7 (p - 250)} (7 (p - 250))$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Move $2$ to the left of $p$ .

$- 21 p + 5250 = 2 \cdot p (7 (p - 250)) + \frac{250}{7 (p - 250)} (7 (p - 250))$

Step 3.3.1.2

Apply the distributive property.

$- 21 p + 5250 = 2 p (7 p + 7 \cdot - 250) + \frac{250}{7 (p - 250)} (7 (p - 250))$

Step 3.3.1.3

Multiply $7$ by $- 250$ .

$- 21 p + 5250 = 2 p (7 p - 1750) + \frac{250}{7 (p - 250)} (7 (p - 250))$

Step 3.3.1.4

Apply the distributive property.

$- 21 p + 5250 = 2 p (7 p) + 2 p \cdot - 1750 + \frac{250}{7 (p - 250)} (7 (p - 250))$

Step 3.3.1.5

Rewrite using the commutative property of multiplication.

$- 21 p + 5250 = 2 \cdot 7 p \cdot p + 2 p \cdot - 1750 + \frac{250}{7 (p - 250)} (7 (p - 250))$

Step 3.3.1.6

Multiply $- 1750$ by $2$ .

$- 21 p + 5250 = 2 \cdot 7 p \cdot p - 3500 p + \frac{250}{7 (p - 250)} (7 (p - 250))$

Step 3.3.1.7

Simplify each term.

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Step 3.3.1.7.1

Multiply $p$ by $p$ by adding the exponents.

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Step 3.3.1.7.1.1

Move $p$ .

$- 21 p + 5250 = 2 \cdot 7 (p \cdot p) - 3500 p + \frac{250}{7 (p - 250)} (7 (p - 250))$

Step 3.3.1.7.1.2

Multiply $p$ by $p$ .

$- 21 p + 5250 = 2 \cdot 7 p^{2} - 3500 p + \frac{250}{7 (p - 250)} (7 (p - 250))$

Step 3.3.1.7.2

Multiply $2$ by $7$ .

$- 21 p + 5250 = 14 p^{2} - 3500 p + \frac{250}{7 (p - 250)} (7 (p - 250))$

Step 3.3.1.8

Rewrite using the commutative property of multiplication.

$- 21 p + 5250 = 14 p^{2} - 3500 p + 7 \frac{250}{7 (p - 250)} (p - 250)$

Step 3.3.1.9

Cancel the common factor of $7$ .

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Step 3.3.1.9.1

Cancel the common factor.

$- 21 p + 5250 = 14 p^{2} - 3500 p + 7 \frac{250}{7 (p - 250)} (p - 250)$

Step 3.3.1.9.2

Rewrite the expression.

$- 21 p + 5250 = 14 p^{2} - 3500 p + \frac{250}{p - 250} (p - 250)$

Step 3.3.1.10

Cancel the common factor of $p - 250$ .

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Step 3.3.1.10.1

Cancel the common factor.

$- 21 p + 5250 = 14 p^{2} - 3500 p + \frac{250}{p - 250} (p - 250)$

Step 3.3.1.10.2

Rewrite the expression.

$- 21 p + 5250 = 14 p^{2} - 3500 p + 250$

Step 4

Solve the equation.

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Step 4.1

Since $p$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$14 p^{2} - 3500 p + 250 = - 21 p + 5250$

Step 4.2

Move all terms containing $p$ to the left side of the equation.

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Step 4.2.1

Add $21 p$ to both sides of the equation.

$14 p^{2} - 3500 p + 250 + 21 p = 5250$

Step 4.2.2

Add $- 3500 p$ and $21 p$ .

$14 p^{2} - 3479 p + 250 = 5250$

Step 4.3

Move all terms to the left side of the equation and simplify.

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Step 4.3.1

Subtract $5250$ from both sides of the equation.

$14 p^{2} - 3479 p + 250 - 5250 = 0$

Step 4.3.2

Subtract $5250$ from $250$ .

$14 p^{2} - 3479 p - 5000 = 0$

Step 4.4

Use the quadratic formula to find the solutions.

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

Step 4.5

Substitute the values $a = 14$ , $b = - 3479$ , and $c = - 5000$ into the quadratic formula and solve for $p$ .

$\frac{3479 \pm \sqrt{{(- 3479)}^{2} - 4 \cdot (14 \cdot - 5000)}}{2 \cdot 14}$

Step 4.6

Simplify.

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Step 4.6.1

Simplify the numerator.

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Step 4.6.1.1

Raise $- 3479$ to the power of $2$ .

$p = \frac{3479 \pm \sqrt{12103441 - 4 \cdot 14 \cdot - 5000}}{2 \cdot 14}$

Step 4.6.1.2

Multiply $- 4 \cdot 14 \cdot - 5000$ .

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Step 4.6.1.2.1

Multiply $- 4$ by $14$ .

$p = \frac{3479 \pm \sqrt{12103441 - 56 \cdot - 5000}}{2 \cdot 14}$

Step 4.6.1.2.2

Multiply $- 56$ by $- 5000$ .

$p = \frac{3479 \pm \sqrt{12103441 + 280000}}{2 \cdot 14}$

Step 4.6.1.3

Add $12103441$ and $280000$ .

$p = \frac{3479 \pm \sqrt{12383441}}{2 \cdot 14}$

Step 4.6.2

Multiply $2$ by $14$ .

$p = \frac{3479 \pm \sqrt{12383441}}{28}$

Step 4.7

The final answer is the combination of both solutions.

$p = \frac{3479 + \sqrt{12383441}}{28}, \frac{3479 - \sqrt{12383441}}{28}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$p = \frac{3479 + \sqrt{12383441}}{28}, \frac{3479 - \sqrt{12383441}}{28}$

Decimal Form:

$p = 249.92897738 \dots, - 1.42897738 \dots$