Precalculus Examples

$x = t + \frac{1}{t}$ , $y = t - \frac{1}{t}$

Step 1

Set up the parametric equation for $x (t)$ to solve the equation for $t$ .

$x = t + \frac{1}{t}$

Step 2

Rewrite the equation as $t + \frac{1}{t} = x$ .

$t + \frac{1}{t} = x$

Step 3

Find the LCD of the terms in the equation.

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Step 3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, t, 1$

Step 3.2

The LCM of one and any expression is the expression.

$t$

Step 4

Multiply each term in $t + \frac{1}{t} = x$ by $t$ to eliminate the fractions.

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Step 4.1

Multiply each term in $t + \frac{1}{t} = x$ by $t$ .

$t \cdot t + \frac{1}{t} t = x t$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $t$ by $t$ .

$t^{2} + \frac{1}{t} t = x t$

Step 4.2.1.2

Cancel the common factor of $t$ .

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Step 4.2.1.2.1

Cancel the common factor.

$t^{2} + \frac{1}{t} t = x t$

Step 4.2.1.2.2

Rewrite the expression.

$t^{2} + 1 = x t$

Step 5

Solve the equation.

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Step 5.1

Subtract $x t$ from both sides of the equation.

$t^{2} + 1 - x t = 0$

Step 5.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5.3

Substitute the values $a = 1$ , $b = - x$ , and $c = 1$ into the quadratic formula and solve for $t$ .

$\frac{x \pm \sqrt{{(- x)}^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

Step 5.4

Simplify.

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Step 5.4.1

Simplify the numerator.

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Step 5.4.1.1

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

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

Step 5.4.1.2

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

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

Step 5.4.1.3

Simplify.

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Step 5.4.1.3.1

Multiply $2 \cdot 1 \cdot 1$ .

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Step 5.4.1.3.1.1

Multiply $2$ by $1$ .

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

Step 5.4.1.3.1.2

Multiply $2$ by $1$ .

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

Step 5.4.1.3.2

Multiply $- (2 \cdot 1 \cdot 1)$ .

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Step 5.4.1.3.2.1

Multiply $2$ by $1$ .

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

Step 5.4.1.3.2.2

Multiply $2$ by $1$ .

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

Step 5.4.1.3.2.3

Multiply $- 1$ by $2$ .

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

Step 5.4.2

Multiply $2$ by $1$ .

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

Step 5.5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 5.5.1

Simplify the numerator.

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Step 5.5.1.1

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

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

Step 5.5.1.2

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

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

Step 5.5.1.3

Simplify.

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Step 5.5.1.3.1

Multiply $2 \cdot 1 \cdot 1$ .

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Step 5.5.1.3.1.1

Multiply $2$ by $1$ .

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

Step 5.5.1.3.1.2

Multiply $2$ by $1$ .

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

Step 5.5.1.3.2

Multiply $- (2 \cdot 1 \cdot 1)$ .

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Step 5.5.1.3.2.1

Multiply $2$ by $1$ .

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

Step 5.5.1.3.2.2

Multiply $2$ by $1$ .

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

Step 5.5.1.3.2.3

Multiply $- 1$ by $2$ .

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

Step 5.5.2

Multiply $2$ by $1$ .

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

Step 5.5.3

Change the $\pm$ to $+$ .

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

Step 5.6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 5.6.1

Simplify the numerator.

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Step 5.6.1.1

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

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

Step 5.6.1.2

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

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

Step 5.6.1.3

Simplify.

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Step 5.6.1.3.1

Multiply $2 \cdot 1 \cdot 1$ .

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Step 5.6.1.3.1.1

Multiply $2$ by $1$ .

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

Step 5.6.1.3.1.2

Multiply $2$ by $1$ .

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

Step 5.6.1.3.2

Multiply $- (2 \cdot 1 \cdot 1)$ .

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Step 5.6.1.3.2.1

Multiply $2$ by $1$ .

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

Step 5.6.1.3.2.2

Multiply $2$ by $1$ .

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

Step 5.6.1.3.2.3

Multiply $- 1$ by $2$ .

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

Step 5.6.2

Multiply $2$ by $1$ .

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

Step 5.6.3

Change the $\pm$ to $-$ .

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

Step 5.7

The final answer is the combination of both solutions.

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

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

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

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

Step 6

Replace $t$ in the equation for $y$ to get the equation in terms of $x$ .

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