Precalculus Examples

x^{2} - 4 y^{2} = 1

$x^{2} - 4 y^{2} = 1$

Step 1

Subtract

x^{2}

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

- 4 y^{2} = 1 - x^{2}

$- 4 y^{2} = 1 - x^{2}$

Step 2

Divide each term in

- 4 y^{2} = 1 - x^{2}

$- 4 y^{2} = 1 - x^{2}$ by

- 4

$- 4$ and simplify.

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Step 2.1

Divide each term in

- 4 y^{2} = 1 - x^{2}

$- 4 y^{2} = 1 - x^{2}$ by

- 4

$- 4$ .

\frac{- 4 y^{2}}{- 4} = \frac{1}{- 4} + \frac{- x^{2}}{- 4}

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of

- 4

$- 4$ .

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Step 2.2.1.1

Cancel the common factor.

\frac{- 4 y^{2}}{- 4} = \frac{1}{- 4} + \frac{- x^{2}}{- 4}

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

Step 2.2.1.2

Divide

y^{2}

$y^{2}$ by

1

$1$ .

y^{2} = \frac{1}{- 4} + \frac{- x^{2}}{- 4}

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

y^{2} = \frac{1}{- 4} + \frac{- x^{2}}{- 4}

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

y^{2} = \frac{1}{- 4} + \frac{- x^{2}}{- 4}

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Move the negative in front of the fraction.

y^{2} = - \frac{1}{4} + \frac{- x^{2}}{- 4}

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

Step 2.3.1.2

Dividing two negative values results in a positive value.

y^{2} = - \frac{1}{4} + \frac{x^{2}}{4}

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

y^{2} = - \frac{1}{4} + \frac{x^{2}}{4}

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

y^{2} = - \frac{1}{4} + \frac{x^{2}}{4}

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

y^{2} = - \frac{1}{4} + \frac{x^{2}}{4}

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

Step 3

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

y = \pm \sqrt{- \frac{1}{4} + \frac{x^{2}}{4}}

$y = \pm \sqrt{- \frac{1}{4} + \frac{x^{2}}{4}}$

Step 4

Simplify

\pm \sqrt{- \frac{1}{4} + \frac{x^{2}}{4}}

$\pm \sqrt{- \frac{1}{4} + \frac{x^{2}}{4}}$ .

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Step 4.1

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

y = \pm \sqrt{- \frac{1}{4} + \frac{x^{2}}{2^{2}}}

$y = \pm \sqrt{- \frac{1}{4} + \frac{x^{2}}{2^{2}}}$

Step 4.2

Rewrite

\frac{x^{2}}{2^{2}}

$\frac{x^{2}}{2^{2}}$ as

{(\frac{x}{2})}^{2}

${(\frac{x}{2})}^{2}$ .

y = \pm \sqrt{- \frac{1}{4} + {(\frac{x}{2})}^{2}}

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

Step 4.3

Simplify the expression.

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Step 4.3.1

Rewrite

\frac{1}{4}

$\frac{1}{4}$ as

{(\frac{1}{2})}^{2}

${(\frac{1}{2})}^{2}$ .

y = \pm \sqrt{- {(\frac{1}{2})}^{2} + {(\frac{x}{2})}^{2}}

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

Step 4.3.2

Reorder

- {(\frac{1}{2})}^{2}

$- {(\frac{1}{2})}^{2}$ and

{(\frac{x}{2})}^{2}

${(\frac{x}{2})}^{2}$ .

y = \pm \sqrt{{(\frac{x}{2})}^{2} - {(\frac{1}{2})}^{2}}

$y = \pm \sqrt{{(\frac{x}{2})}^{2} - {(\frac{1}{2})}^{2}}$

y = \pm \sqrt{{(\frac{x}{2})}^{2} - {(\frac{1}{2})}^{2}}

$y = \pm \sqrt{{(\frac{x}{2})}^{2} - {(\frac{1}{2})}^{2}}$

Step 4.4

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = \frac{x}{2}

$a = \frac{x}{2}$ and

b = \frac{1}{2}

$b = \frac{1}{2}$ .

y = \pm \sqrt{(\frac{x}{2} + \frac{1}{2}) (\frac{x}{2} - \frac{1}{2})}

$y = \pm \sqrt{(\frac{x}{2} + \frac{1}{2}) (\frac{x}{2} - \frac{1}{2})}$

Step 4.5

Simplify terms.

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Step 4.5.1

Combine the numerators over the common denominator.

y = \pm \sqrt{\frac{x + 1}{2} (\frac{x}{2} - \frac{1}{2})}

$y = \pm \sqrt{\frac{x + 1}{2} (\frac{x}{2} - \frac{1}{2})}$

Step 4.5.2

Combine the numerators over the common denominator.

y = \pm \sqrt{\frac{x + 1}{2} \cdot \frac{x - 1}{2}}

$y = \pm \sqrt{\frac{x + 1}{2} \cdot \frac{x - 1}{2}}$

Step 4.5.3

Multiply

\frac{x + 1}{2}

$\frac{x + 1}{2}$ by

\frac{x - 1}{2}

$\frac{x - 1}{2}$ .

y = \pm \sqrt{\frac{(x + 1) (x - 1)}{2 \cdot 2}}

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

Step 4.5.4

Multiply

2

$2$ by

2

$2$ .

y = \pm \sqrt{\frac{(x + 1) (x - 1)}{4}}

$y = \pm \sqrt{\frac{(x + 1) (x - 1)}{4}}$

y = \pm \sqrt{\frac{(x + 1) (x - 1)}{4}}

$y = \pm \sqrt{\frac{(x + 1) (x - 1)}{4}}$

Step 4.6

Rewrite

\frac{(x + 1) (x - 1)}{4}

$\frac{(x + 1) (x - 1)}{4}$ as

{(\frac{1}{2})}^{2} ((x + 1) (x - 1))

${(\frac{1}{2})}^{2} ((x + 1) (x - 1))$ .

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Step 4.6.1

Factor the perfect power

1^{2}

$1^{2}$ out of

(x + 1) (x - 1)

$(x + 1) (x - 1)$ .

y = \pm \sqrt{\frac{1^{2} ((x + 1) (x - 1))}{4}}

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

Step 4.6.2

Factor the perfect power

2^{2}

$2^{2}$ out of

4

$4$ .

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

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

Step 4.6.3

Rearrange the fraction

\frac{1^{2} ((x + 1) (x - 1))}{2^{2} \cdot 1}

$\frac{1^{2} ((x + 1) (x - 1))}{2^{2} \cdot 1}$ .

y = \pm \sqrt{{(\frac{1}{2})}^{2} ((x + 1) (x - 1))}

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

y = \pm \sqrt{{(\frac{1}{2})}^{2} ((x + 1) (x - 1))}

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

Step 4.7

Pull terms out from under the radical.

y = \pm \frac{1}{2} \sqrt{(x + 1) (x - 1)}

$y = \pm \frac{1}{2} \sqrt{(x + 1) (x - 1)}$

Step 4.8

Combine

\frac{1}{2}

$\frac{1}{2}$ and

\sqrt{(x + 1) (x - 1)}

$\sqrt{(x + 1) (x - 1)}$ .

y = \pm \frac{\sqrt{(x + 1) (x - 1)}}{2}

$y = \pm \frac{\sqrt{(x + 1) (x - 1)}}{2}$

y = \pm \frac{\sqrt{(x + 1) (x - 1)}}{2}

$y = \pm \frac{\sqrt{(x + 1) (x - 1)}}{2}$

Step 5

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

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Step 5.1

First, use the positive value of the

\pm

$\pm$ to find the first solution.

y = \frac{\sqrt{(x + 1) (x - 1)}}{2}

$y = \frac{\sqrt{(x + 1) (x - 1)}}{2}$

Step 5.2

Next, use the negative value of the

\pm

$\pm$ to find the second solution.

y = - \frac{\sqrt{(x + 1) (x - 1)}}{2}

$y = - \frac{\sqrt{(x + 1) (x - 1)}}{2}$

Step 5.3

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

y = \frac{\sqrt{(x + 1) (x - 1)}}{2}

$y = \frac{\sqrt{(x + 1) (x - 1)}}{2}$

y = - \frac{\sqrt{(x + 1) (x - 1)}}{2}

$y = - \frac{\sqrt{(x + 1) (x - 1)}}{2}$

y = \frac{\sqrt{(x + 1) (x - 1)}}{2}

$y = \frac{\sqrt{(x + 1) (x - 1)}}{2}$

y = - \frac{\sqrt{(x + 1) (x - 1)}}{2}

$y = - \frac{\sqrt{(x + 1) (x - 1)}}{2}$

Step 6

To rewrite as a function of

x

$x$ , write the equation so that

y

$y$ is by itself on one side of the equal sign and an expression involving only

x

$x$ is on the other side.

f (x) = \frac{\sqrt{(x + 1) (x - 1)}}{2}, - \frac{\sqrt{(x + 1) (x - 1)}}{2}

$f (x) = \frac{\sqrt{(x + 1) (x - 1)}}{2}, - \frac{\sqrt{(x + 1) (x - 1)}}{2}$