Finite Math Examples

$\frac{{(y + 12)}^{2}}{9} - \frac{x^{2}}{4} = 1$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$\frac{{((0) + 12)}^{2}}{9} - \frac{x^{2}}{4} = 1$

Step 1.2

Solve the equation.

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Step 1.2.1

Simplify $\frac{{((0) + 12)}^{2}}{9} - \frac{x^{2}}{4}$ .

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Step 1.2.1.1

Add $0$ and $12$ .

$\frac{12^{2}}{9} - \frac{x^{2}}{4} = 1$

Step 1.2.1.2

Simplify each term.

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Step 1.2.1.2.1

Raise $12$ to the power of $2$ .

$\frac{144}{9} - \frac{x^{2}}{4} = 1$

Step 1.2.1.2.2

Divide $144$ by $9$ .

$16 - \frac{x^{2}}{4} = 1$

Step 1.2.2

Move all terms not containing $x$ to the right side of the equation.

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Step 1.2.2.1

Subtract $16$ from both sides of the equation.

$- \frac{x^{2}}{4} = 1 - 16$

Step 1.2.2.2

Subtract $16$ from $1$ .

$- \frac{x^{2}}{4} = - 15$

Step 1.2.3

Multiply both sides of the equation by $- 4$ .

$- 4 (- \frac{x^{2}}{4}) = - 4 \cdot - 15$

Step 1.2.4

Simplify both sides of the equation.

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Step 1.2.4.1

Simplify the left side.

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Step 1.2.4.1.1

Simplify $- 4 (- \frac{x^{2}}{4})$ .

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Step 1.2.4.1.1.1

Cancel the common factor of $4$ .

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Step 1.2.4.1.1.1.1

Move the leading negative in $- \frac{x^{2}}{4}$ into the numerator.

$- 4 \frac{- x^{2}}{4} = - 4 \cdot - 15$

Step 1.2.4.1.1.1.2

Factor $4$ out of $- 4$ .

$4 (- 1) \frac{- x^{2}}{4} = - 4 \cdot - 15$

Step 1.2.4.1.1.1.3

Cancel the common factor.

$4 \cdot - 1 \frac{- x^{2}}{4} = - 4 \cdot - 15$

Step 1.2.4.1.1.1.4

Rewrite the expression.

$- - x^{2} = - 4 \cdot - 15$

Step 1.2.4.1.1.2

Multiply.

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Step 1.2.4.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 x^{2} = - 4 \cdot - 15$

Step 1.2.4.1.1.2.2

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

$x^{2} = - 4 \cdot - 15$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Multiply $- 4$ by $- 15$ .

$x^{2} = 60$

Step 1.2.5

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

$x = \pm \sqrt{60}$

Step 1.2.6

Simplify $\pm \sqrt{60}$ .

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Step 1.2.6.1

Rewrite $60$ as $2^{2} \cdot 15$ .

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Step 1.2.6.1.1

Factor $4$ out of $60$ .

$x = \pm \sqrt{4 (15)}$

Step 1.2.6.1.2

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

$x = \pm \sqrt{2^{2} \cdot 15}$

Step 1.2.6.2

Pull terms out from under the radical.

$x = \pm 2 \sqrt{15}$

Step 1.2.7

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

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Step 1.2.7.1

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

$x = 2 \sqrt{15}$

Step 1.2.7.2

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

$x = - 2 \sqrt{15}$

Step 1.2.7.3

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

$x = 2 \sqrt{15}, - 2 \sqrt{15}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(2 \sqrt{15}, 0), (- 2 \sqrt{15}, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$\frac{{(y + 12)}^{2}}{9} - \frac{{(0)}^{2}}{4} = 1$

Step 2.2

Solve the equation.

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Step 2.2.1

Add $\frac{{(0)}^{2}}{4}$ to both sides of the equation.

$\frac{{(y + 12)}^{2}}{9} = 1 + \frac{{(0)}^{2}}{4}$

Step 2.2.2

Simplify $1 + \frac{{(0)}^{2}}{4}$ .

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{{(y + 12)}^{2}}{9} = 1 + \frac{0}{4}$

Step 2.2.2.1.2

Divide $0$ by $4$ .

$\frac{{(y + 12)}^{2}}{9} = 1 + 0$

Step 2.2.2.2

Add $1$ and $0$ .

$\frac{{(y + 12)}^{2}}{9} = 1$

Step 2.2.3

Multiply both sides of the equation by $9$ .

$9 \frac{{(y + 12)}^{2}}{9} = 9 \cdot 1$

Step 2.2.4

Simplify both sides of the equation.

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Step 2.2.4.1

Simplify the left side.

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Step 2.2.4.1.1

Cancel the common factor of $9$ .

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Step 2.2.4.1.1.1

Cancel the common factor.

$9 \frac{{(y + 12)}^{2}}{9} = 9 \cdot 1$

Step 2.2.4.1.1.2

Rewrite the expression.

${(y + 12)}^{2} = 9 \cdot 1$

Step 2.2.4.2

Simplify the right side.

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Step 2.2.4.2.1

Multiply $9$ by $1$ .

${(y + 12)}^{2} = 9$

Step 2.2.5

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

$y + 12 = \pm \sqrt{9}$

Step 2.2.6

Simplify $\pm \sqrt{9}$ .

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Step 2.2.6.1

Rewrite $9$ as $3^{2}$ .

$y + 12 = \pm \sqrt{3^{2}}$

Step 2.2.6.2

Pull terms out from under the radical, assuming positive real numbers.

$y + 12 = \pm 3$

Step 2.2.7

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

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Step 2.2.7.1

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

$y + 12 = 3$

Step 2.2.7.2

Move all terms not containing $y$ to the right side of the equation.

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Step 2.2.7.2.1

Subtract $12$ from both sides of the equation.

$y = 3 - 12$

Step 2.2.7.2.2

Subtract $12$ from $3$ .

$y = - 9$

Step 2.2.7.3

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

$y + 12 = - 3$

Step 2.2.7.4

Move all terms not containing $y$ to the right side of the equation.

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Step 2.2.7.4.1

Subtract $12$ from both sides of the equation.

$y = - 3 - 12$

Step 2.2.7.4.2

Subtract $12$ from $- 3$ .

$y = - 15$

Step 2.2.7.5

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

$y = - 9, - 15$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 9), (0, - 15)$

Step 3

List the intersections.

x-intercept(s): $(2 \sqrt{15}, 0), (- 2 \sqrt{15}, 0)$

y-intercept(s): $(0, - 9), (0, - 15)$

Step 4