Precalculus Examples

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

Step 1

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

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

Step 2

Divide each term in $- 4 y^{2} = 1 - x^{2}$ by $- 4$ and simplify.

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Step 2.1

Divide each term in $- 4 y^{2} = 1 - x^{2}$ by $- 4$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $- 4$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

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

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

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.

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

Step 2.3.1.2

Dividing two negative values results in a positive value.

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Simplify the expression.

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Step 4.3.1

Rewrite $\frac{1}{4}$ as ${(\frac{1}{2})}^{2}$ .

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

Step 4.3.2

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

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

Step 4.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{x}{2}$ and $b = \frac{1}{2}$ .

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

Step 4.5

Simplify terms.

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Step 4.5.1

Combine the numerators over the common denominator.

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

Step 4.5.2

Combine the numerators over the common denominator.

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

Step 4.5.3

Multiply $\frac{x + 1}{2}$ by $\frac{x - 1}{2}$ .

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

Step 4.5.4

Multiply $2$ by $2$ .

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

Step 4.6

Rewrite $\frac{(x + 1) (x - 1)}{4}$ as ${(\frac{1}{2})}^{2} ((x + 1) (x - 1))$ .

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Step 4.6.1

Factor the perfect power $1^{2}$ out of $(x + 1) (x - 1)$ .

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

Step 4.6.2

Factor the perfect power $2^{2}$ out of $4$ .

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

Step 4.6.3

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

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

Step 4.7

Pull terms out from under the radical.

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

Step 4.8

Combine $\frac{1}{2}$ and $\sqrt{(x + 1) (x - 1)}$ .

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

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 $\pm$ to find the first solution.

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

Step 5.2

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

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

Step 5.3

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

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

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

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

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

Step 6

To rewrite as a function of $x$ , write the equation so that $y$ is by itself on one side of the equal sign and an expression involving only $x$ is on the other side.

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