Finite Math Examples

$y = \frac{25 - x^{2}}{8 + x^{2}}$

Step 1

Rewrite the equation as $\frac{25 - x^{2}}{8 + x^{2}} = y$ .

$\frac{25 - x^{2}}{8 + x^{2}} = y$

Step 2

Factor each term.

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Step 2.1

Rewrite $25$ as $5^{2}$ .

$\frac{5^{2} - x^{2}}{8 + x^{2}} = y$

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = x$ .

$\frac{(5 + x) (5 - x)}{8 + x^{2}} = y$

Step 3

Find the LCD of the terms in the equation.

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Step 3.1

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

$8 + x^{2}, 1$

Step 3.2

Remove parentheses.

$8 + x^{2}, 1$

Step 3.3

The LCM of one and any expression is the expression.

$8 + x^{2}$

Step 4

Multiply each term in $\frac{(5 + x) (5 - x)}{8 + x^{2}} = y$ by $8 + x^{2}$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{(5 + x) (5 - x)}{8 + x^{2}} = y$ by $8 + x^{2}$ .

$\frac{(5 + x) (5 - x)}{8 + x^{2}} (8 + x^{2}) = y (8 + x^{2})$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $8 + x^{2}$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{(5 + x) (5 - x)}{8 + x^{2}} (8 + x^{2}) = y (8 + x^{2})$

Step 4.2.1.2

Rewrite the expression.

$(5 + x) (5 - x) = y (8 + x^{2})$

Step 4.2.2

Expand $(5 + x) (5 - x)$ using the FOIL Method.

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Step 4.2.2.1

Apply the distributive property.

$5 (5 - x) + x (5 - x) = y (8 + x^{2})$

Step 4.2.2.2

Apply the distributive property.

$5 \cdot 5 + 5 (- x) + x (5 - x) = y (8 + x^{2})$

Step 4.2.2.3

Apply the distributive property.

$5 \cdot 5 + 5 (- x) + x \cdot 5 + x (- x) = y (8 + x^{2})$

Step 4.2.3

Simplify and combine like terms.

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Step 4.2.3.1

Simplify each term.

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Step 4.2.3.1.1

Multiply $5$ by $5$ .

$25 + 5 (- x) + x \cdot 5 + x (- x) = y (8 + x^{2})$

Step 4.2.3.1.2

Multiply $- 1$ by $5$ .

$25 - 5 x + x \cdot 5 + x (- x) = y (8 + x^{2})$

Step 4.2.3.1.3

Move $5$ to the left of $x$ .

$25 - 5 x + 5 \cdot x + x (- x) = y (8 + x^{2})$

Step 4.2.3.1.4

Rewrite using the commutative property of multiplication.

$25 - 5 x + 5 x - x \cdot x = y (8 + x^{2})$

Step 4.2.3.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 4.2.3.1.5.1

Move $x$ .

$25 - 5 x + 5 x - (x \cdot x) = y (8 + x^{2})$

Step 4.2.3.1.5.2

Multiply $x$ by $x$ .

$25 - 5 x + 5 x - x^{2} = y (8 + x^{2})$

Step 4.2.3.2

Add $- 5 x$ and $5 x$ .

$25 + 0 - x^{2} = y (8 + x^{2})$

Step 4.2.3.3

Add $25$ and $0$ .

$25 - x^{2} = y (8 + x^{2})$

Step 4.3

Simplify the right side.

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Step 4.3.1

Apply the distributive property.

$25 - x^{2} = y \cdot 8 + y x^{2}$

Step 4.3.2

Move $8$ to the left of $y$ .

$25 - x^{2} = 8 y + y x^{2}$

Step 5

Solve the equation.

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Step 5.1

Subtract $y x^{2}$ from both sides of the equation.

$25 - x^{2} - y x^{2} = 8 y$

Step 5.2

Subtract $25$ from both sides of the equation.

$- x^{2} - y x^{2} = 8 y - 25$

Step 5.3

Factor $x^{2}$ out of $- x^{2} - y x^{2}$ .

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Step 5.3.1

Factor $x^{2}$ out of $- x^{2}$ .

$x^{2} \cdot - 1 - y x^{2} = 8 y - 25$

Step 5.3.2

Factor $x^{2}$ out of $- y x^{2}$ .

$x^{2} \cdot - 1 + x^{2} (- y) = 8 y - 25$

Step 5.3.3

Factor $x^{2}$ out of $x^{2} \cdot - 1 + x^{2} (- y)$ .

$x^{2} (- 1 - y) = 8 y - 25$

Step 5.4

Divide each term in $x^{2} (- 1 - y) = 8 y - 25$ by $- 1 - y$ and simplify.

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Step 5.4.1

Divide each term in $x^{2} (- 1 - y) = 8 y - 25$ by $- 1 - y$ .

$\frac{x^{2} (- 1 - y)}{- 1 - y} = \frac{8 y}{- 1 - y} + \frac{- 25}{- 1 - y}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $- 1 - y$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{x^{2} (- 1 - y)}{- 1 - y} = \frac{8 y}{- 1 - y} + \frac{- 25}{- 1 - y}$

Step 5.4.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{8 y}{- 1 - y} + \frac{- 25}{- 1 - y}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Combine the numerators over the common denominator.

$x^{2} = \frac{8 y - 25}{- 1 - y}$

Step 5.4.3.2

Rewrite $- 1$ as $- 1 (1)$ .

$x^{2} = \frac{8 y - 25}{- 1 (1) - y}$

Step 5.4.3.3

Factor $- 1$ out of $- y$ .

$x^{2} = \frac{8 y - 25}{- 1 (1) - (y)}$

Step 5.4.3.4

Factor $- 1$ out of $- 1 (1) - (y)$ .

$x^{2} = \frac{8 y - 25}{- 1 (1 + y)}$

Step 5.4.3.5

Move the negative in front of the fraction.

$x^{2} = - \frac{8 y - 25}{1 + y}$

Step 5.5

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- \frac{8 y - 25}{1 + y}}$

Step 5.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.6.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{- \frac{8 y - 25}{1 + y}}$

Step 5.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{- \frac{8 y - 25}{1 + y}}$

Step 5.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{- \frac{8 y - 25}{1 + y}}$

$x = - \sqrt{- \frac{8 y - 25}{1 + y}}$

$x = \sqrt{- \frac{8 y - 25}{1 + y}}$

$x = - \sqrt{- \frac{8 y - 25}{1 + y}}$

$x = \sqrt{- \frac{8 y - 25}{1 + y}}$

$x = - \sqrt{- \frac{8 y - 25}{1 + y}}$