Precalculus Examples

$\frac{y^{2}}{12} - \frac{x^{2}}{8} = 1$

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 $x$ .

$\frac{{(0)}^{2}}{12} - \frac{x^{2}}{8} = 1$

Step 1.2

Solve the equation.

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Step 1.2.1

Simplify $\frac{{(0)}^{2}}{12} - \frac{x^{2}}{8}$ .

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

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

$\frac{0}{12} - \frac{x^{2}}{8} = 1$

Step 1.2.1.1.2

Divide $0$ by $12$ .

$0 - \frac{x^{2}}{8} = 1$

Step 1.2.1.2

Subtract $\frac{x^{2}}{8}$ from $0$ .

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

Step 1.2.2

Multiply both sides of the equation by $- 8$ .

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

Step 1.2.3

Simplify both sides of the equation.

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Step 1.2.3.1

Simplify the left side.

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Step 1.2.3.1.1

Simplify $- 8 (- \frac{x^{2}}{8})$ .

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Step 1.2.3.1.1.1

Cancel the common factor of $8$ .

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Step 1.2.3.1.1.1.1

Move the leading negative in $- \frac{x^{2}}{8}$ into the numerator.

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

Step 1.2.3.1.1.1.2

Factor $8$ out of $- 8$ .

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

Step 1.2.3.1.1.1.3

Cancel the common factor.

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

Step 1.2.3.1.1.1.4

Rewrite the expression.

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

Step 1.2.3.1.1.2

Multiply.

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Step 1.2.3.1.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 1.2.3.1.1.2.2

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

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

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply $- 8$ by $1$ .

$x^{2} = - 8$

Step 1.2.4

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

$x = \pm \sqrt{- 8}$

Step 1.2.5

Simplify $\pm \sqrt{- 8}$ .

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Step 1.2.5.1

Rewrite $- 8$ as $- 1 (8)$ .

$x = \pm \sqrt{- 1 (8)}$

Step 1.2.5.2

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{8}$

Step 1.2.5.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \cdot \sqrt{8}$

Step 1.2.5.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 1.2.5.4.1

Factor $4$ out of $8$ .

$x = \pm i \cdot \sqrt{4 (2)}$

Step 1.2.5.4.2

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

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

Step 1.2.5.5

Pull terms out from under the radical.

$x = \pm i \cdot (2 \sqrt{2})$

Step 1.2.5.6

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

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

Step 1.2.6

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

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Step 1.2.6.1

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

$x = 2 i \sqrt{2}$

Step 1.2.6.2

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

$x = - 2 i \sqrt{2}$

Step 1.2.6.3

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

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

Step 1.3

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

x-intercept(s): $None$

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 $y$ .

$\frac{y^{2}}{12} - \frac{{(0)}^{2}}{8} = 1$

Step 2.2

Solve the equation.

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Step 2.2.1

Simplify $\frac{y^{2}}{12} - \frac{{(0)}^{2}}{8}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

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

$\frac{y^{2}}{12} - \frac{0}{8} = 1$

Step 2.2.1.1.2

Divide $0$ by $8$ .

$\frac{y^{2}}{12} - 0 = 1$

Step 2.2.1.1.3

Multiply $- 1$ by $0$ .

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

Step 2.2.1.2

Add $\frac{y^{2}}{12}$ and $0$ .

$\frac{y^{2}}{12} = 1$

Step 2.2.2

Multiply both sides of the equation by $12$ .

$12 \frac{y^{2}}{12} = 12 \cdot 1$

Step 2.2.3

Simplify both sides of the equation.

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Step 2.2.3.1

Simplify the left side.

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Step 2.2.3.1.1

Cancel the common factor of $12$ .

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Step 2.2.3.1.1.1

Cancel the common factor.

$12 \frac{y^{2}}{12} = 12 \cdot 1$

Step 2.2.3.1.1.2

Rewrite the expression.

$y^{2} = 12 \cdot 1$

Step 2.2.3.2

Simplify the right side.

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Step 2.2.3.2.1

Multiply $12$ by $1$ .

$y^{2} = 12$

Step 2.2.4

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

$y = \pm \sqrt{12}$

Step 2.2.5

Simplify $\pm \sqrt{12}$ .

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Step 2.2.5.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.2.5.1.1

Factor $4$ out of $12$ .

$y = \pm \sqrt{4 (3)}$

Step 2.2.5.1.2

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

$y = \pm \sqrt{2^{2} \cdot 3}$

Step 2.2.5.2

Pull terms out from under the radical.

$y = \pm 2 \sqrt{3}$

Step 2.2.6

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

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Step 2.2.6.1

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

$y = 2 \sqrt{3}$

Step 2.2.6.2

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

$y = - 2 \sqrt{3}$

Step 2.2.6.3

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

$y = 2 \sqrt{3}, - 2 \sqrt{3}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 2 \sqrt{3}), (0, - 2 \sqrt{3})$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, 2 \sqrt{3}), (0, - 2 \sqrt{3})$

Step 4