Finite Math Examples

$\frac{x^{2}}{225} + \frac{y^{2}}{625} = 1$

Step 1

Subtract $\frac{x^{2}}{225}$ from both sides of the equation.

$\frac{y^{2}}{625} = 1 - \frac{x^{2}}{225}$

Step 2

Multiply both sides of the equation by $625$ .

$625 \frac{y^{2}}{625} = 625 (1 - \frac{x^{2}}{225})$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $625$ .

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Step 3.1.1.1

Cancel the common factor.

$625 \frac{y^{2}}{625} = 625 (1 - \frac{x^{2}}{225})$

Step 3.1.1.2

Rewrite the expression.

$y^{2} = 625 (1 - \frac{x^{2}}{225})$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $625 (1 - \frac{x^{2}}{225})$ .

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Step 3.2.1.1

Apply the distributive property.

$y^{2} = 625 \cdot 1 + 625 (- \frac{x^{2}}{225})$

Step 3.2.1.2

Multiply $625$ by $1$ .

$y^{2} = 625 + 625 (- \frac{x^{2}}{225})$

Step 3.2.1.3

Cancel the common factor of $25$ .

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Step 3.2.1.3.1

Move the leading negative in $- \frac{x^{2}}{225}$ into the numerator.

$y^{2} = 625 + 625 \frac{- x^{2}}{225}$

Step 3.2.1.3.2

Factor $25$ out of $625$ .

$y^{2} = 625 + 25 (25) \frac{- x^{2}}{225}$

Step 3.2.1.3.3

Factor $25$ out of $225$ .

$y^{2} = 625 + 25 \cdot 25 \frac{- x^{2}}{25 \cdot 9}$

Step 3.2.1.3.4

Cancel the common factor.

$y^{2} = 625 + 25 \cdot 25 \frac{- x^{2}}{25 \cdot 9}$

Step 3.2.1.3.5

Rewrite the expression.

$y^{2} = 625 + 25 \frac{- x^{2}}{9}$

Step 3.2.1.4

Combine $25$ and $\frac{- x^{2}}{9}$ .

$y^{2} = 625 + \frac{25 (- x^{2})}{9}$

Step 3.2.1.5

Simplify the expression.

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Step 3.2.1.5.1

Multiply $- 1$ by $25$ .

$y^{2} = 625 + \frac{- 25 x^{2}}{9}$

Step 3.2.1.5.2

Move the negative in front of the fraction.

$y^{2} = 625 - \frac{25 x^{2}}{9}$

Step 4

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

$y = \pm \sqrt{625 - \frac{25 x^{2}}{9}}$

Step 5

Simplify $\pm \sqrt{625 - \frac{25 x^{2}}{9}}$ .

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Step 5.1

Write the expression using exponents.

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Step 5.1.1

Rewrite $625$ as $25^{2}$ .

$y = \pm \sqrt{25^{2} - \frac{25 x^{2}}{9}}$

Step 5.1.2

Rewrite $\frac{25 x^{2}}{9}$ as ${(\frac{5 x}{3})}^{2}$ .

$y = \pm \sqrt{25^{2} - {(\frac{5 x}{3})}^{2}}$

Step 5.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 25$ and $b = \frac{5 x}{3}$ .

$y = \pm \sqrt{(25 + \frac{5 x}{3}) (25 - \frac{5 x}{3})}$

Step 5.3

To write $25$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \pm \sqrt{(25 \cdot \frac{3}{3} + \frac{5 x}{3}) (25 - \frac{5 x}{3})}$

Step 5.4

Simplify terms.

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Step 5.4.1

Combine $25$ and $\frac{3}{3}$ .

$y = \pm \sqrt{(\frac{25 \cdot 3}{3} + \frac{5 x}{3}) (25 - \frac{5 x}{3})}$

Step 5.4.2

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{25 \cdot 3 + 5 x}{3} (25 - \frac{5 x}{3})}$

Step 5.5

Simplify the numerator.

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Step 5.5.1

Factor $5$ out of $25 \cdot 3 + 5 x$ .

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Step 5.5.1.1

Factor $5$ out of $25 \cdot 3$ .

$y = \pm \sqrt{\frac{5 (5 \cdot 3) + 5 x}{3} (25 - \frac{5 x}{3})}$

Step 5.5.1.2

Factor $5$ out of $5 x$ .

$y = \pm \sqrt{\frac{5 (5 \cdot 3) + 5 (x)}{3} (25 - \frac{5 x}{3})}$

Step 5.5.1.3

Factor $5$ out of $5 (5 \cdot 3) + 5 (x)$ .

$y = \pm \sqrt{\frac{5 (5 \cdot 3 + x)}{3} (25 - \frac{5 x}{3})}$

Step 5.5.2

Multiply $5$ by $3$ .

$y = \pm \sqrt{\frac{5 (15 + x)}{3} (25 - \frac{5 x}{3})}$

Step 5.6

To write $25$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \pm \sqrt{\frac{5 (15 + x)}{3} (25 \cdot \frac{3}{3} - \frac{5 x}{3})}$

Step 5.7

Simplify terms.

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Step 5.7.1

Combine $25$ and $\frac{3}{3}$ .

$y = \pm \sqrt{\frac{5 (15 + x)}{3} (\frac{25 \cdot 3}{3} - \frac{5 x}{3})}$

Step 5.7.2

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{5 (15 + x)}{3} \cdot \frac{25 \cdot 3 - 5 x}{3}}$

Step 5.8

Simplify the numerator.

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Step 5.8.1

Factor $5$ out of $25 \cdot 3 - 5 x$ .

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Step 5.8.1.1

Factor $5$ out of $25 \cdot 3$ .

$y = \pm \sqrt{\frac{5 (15 + x)}{3} \cdot \frac{5 (5 \cdot 3) - 5 x}{3}}$

Step 5.8.1.2

Factor $5$ out of $- 5 x$ .

$y = \pm \sqrt{\frac{5 (15 + x)}{3} \cdot \frac{5 (5 \cdot 3) + 5 (- x)}{3}}$

Step 5.8.1.3

Factor $5$ out of $5 (5 \cdot 3) + 5 (- x)$ .

$y = \pm \sqrt{\frac{5 (15 + x)}{3} \cdot \frac{5 (5 \cdot 3 - x)}{3}}$

Step 5.8.2

Multiply $5$ by $3$ .

$y = \pm \sqrt{\frac{5 (15 + x)}{3} \cdot \frac{5 (15 - x)}{3}}$

Step 5.9

Combine fractions.

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Step 5.9.1

Multiply $\frac{5 (15 + x)}{3}$ by $\frac{5 (15 - x)}{3}$ .

$y = \pm \sqrt{\frac{5 (15 + x) (5 (15 - x))}{3 \cdot 3}}$

Step 5.9.2

Multiply.

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Step 5.9.2.1

Multiply $5$ by $5$ .

$y = \pm \sqrt{\frac{25 (15 + x) (15 - x)}{3 \cdot 3}}$

Step 5.9.2.2

Multiply $3$ by $3$ .

$y = \pm \sqrt{\frac{25 (15 + x) (15 - x)}{9}}$

Step 5.10

Rewrite $\frac{25 (15 + x) (15 - x)}{9}$ as ${(\frac{5}{3})}^{2} ((15 + x) (15 - x))$ .

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Step 5.10.1

Factor the perfect power $5^{2}$ out of $25 (15 + x) (15 - x)$ .

$y = \pm \sqrt{\frac{5^{2} ((15 + x) (15 - x))}{9}}$

Step 5.10.2

Factor the perfect power $3^{2}$ out of $9$ .

$y = \pm \sqrt{\frac{5^{2} ((15 + x) (15 - x))}{3^{2} \cdot 1}}$

Step 5.10.3

Rearrange the fraction $\frac{5^{2} ((15 + x) (15 - x))}{3^{2} \cdot 1}$ .

$y = \pm \sqrt{{(\frac{5}{3})}^{2} ((15 + x) (15 - x))}$

Step 5.11

Pull terms out from under the radical.

$y = \pm \frac{5}{3} \sqrt{(15 + x) (15 - x)}$

Step 5.12

Combine $\frac{5}{3}$ and $\sqrt{(15 + x) (15 - x)}$ .

$y = \pm \frac{5 \sqrt{(15 + x) (15 - x)}}{3}$

Step 6

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

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Step 6.1

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

$y = \frac{5 \sqrt{(15 + x) (15 - x)}}{3}$

Step 6.2

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

$y = - \frac{5 \sqrt{(15 + x) (15 - x)}}{3}$

Step 6.3

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

$y = \frac{5 \sqrt{(15 + x) (15 - x)}}{3}$

$y = - \frac{5 \sqrt{(15 + x) (15 - x)}}{3}$

$y = \frac{5 \sqrt{(15 + x) (15 - x)}}{3}$

$y = - \frac{5 \sqrt{(15 + x) (15 - x)}}{3}$

Step 7

Set the radicand in $\sqrt{(15 + x) (15 - x)}$ greater than or equal to $0$ to find where the expression is defined.

$(15 + x) (15 - x) \geq 0$

Step 8

Solve for $x$ .

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Step 8.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$15 + x = 0$

$15 - x = 0$

Step 8.2

Set $15 + x$ equal to $0$ and solve for $x$ .

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Step 8.2.1

Set $15 + x$ equal to $0$ .

$15 + x = 0$

Step 8.2.2

Subtract $15$ from both sides of the equation.

$x = - 15$

Step 8.3

Set $15 - x$ equal to $0$ and solve for $x$ .

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Step 8.3.1

Set $15 - x$ equal to $0$ .

$15 - x = 0$

Step 8.3.2

Solve $15 - x = 0$ for $x$ .

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Step 8.3.2.1

Subtract $15$ from both sides of the equation.

$- x = - 15$

Step 8.3.2.2

Divide each term in $- x = - 15$ by $- 1$ and simplify.

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Step 8.3.2.2.1

Divide each term in $- x = - 15$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 15}{- 1}$

Step 8.3.2.2.2

Simplify the left side.

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Step 8.3.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 15}{- 1}$

Step 8.3.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 15}{- 1}$

Step 8.3.2.2.3

Simplify the right side.

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Step 8.3.2.2.3.1

Divide $- 15$ by $- 1$ .

$x = 15$

Step 8.4

The final solution is all the values that make $(15 + x) (15 - x) \geq 0$ true.

$x = - 15, 15$

Step 8.5

Use each root to create test intervals.

$x < - 15$

$- 15 < x < 15$

$x > 15$

Step 8.6

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 8.6.1

Test a value on the interval $x < - 15$ to see if it makes the inequality true.

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Step 8.6.1.1

Choose a value on the interval $x < - 15$ and see if this value makes the original inequality true.

$x = - 18$

Step 8.6.1.2

Replace $x$ with $- 18$ in the original inequality.

$(15 - 18) (15 - (- 18)) \geq 0$

Step 8.6.1.3

The left side $- 99$ is less than the right side $0$ , which means that the given statement is false.

False

Step 8.6.2

Test a value on the interval $- 15 < x < 15$ to see if it makes the inequality true.

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Step 8.6.2.1

Choose a value on the interval $- 15 < x < 15$ and see if this value makes the original inequality true.

$x = 0$

Step 8.6.2.2

Replace $x$ with $0$ in the original inequality.

$(15 + 0) (15 - (0)) \geq 0$

Step 8.6.2.3

The left side $225$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 8.6.3

Test a value on the interval $x > 15$ to see if it makes the inequality true.

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Step 8.6.3.1

Choose a value on the interval $x > 15$ and see if this value makes the original inequality true.

$x = 18$

Step 8.6.3.2

Replace $x$ with $18$ in the original inequality.

$(15 + 18) (15 - (18)) \geq 0$

Step 8.6.3.3

The left side $- 99$ is less than the right side $0$ , which means that the given statement is false.

False

Step 8.6.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 15$ False

$- 15 < x < 15$ True

$x > 15$ False

$x < - 15$ False

$- 15 < x < 15$ True

$x > 15$ False

Step 8.7

The solution consists of all of the true intervals.

$- 15 \leq x \leq 15$

Step 9

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$[- 15, 15]$

Set-Builder Notation:

${x | - 15 \leq x \leq 15}$

Step 10

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[- 25, 25]$

Set-Builder Notation:

${y | - 25 \leq y \leq 25}$

Step 11

Determine the domain and range.

Domain: $[- 15, 15], {x | - 15 \leq x \leq 15}$

Range: $[- 25, 25], {y | - 25 \leq y \leq 25}$

Step 12