Precalculus Examples

$x = 4 \sqrt{4 - t^{2}}$ , $y = 3 (1 - \sqrt{4 - t^{2}})$

Step 1

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

$x = 4 \sqrt{4 - t^{2}}$

Step 2

Rewrite the equation as $4 \sqrt{4 - t^{2}} = x$ .

$4 \sqrt{4 - t^{2}} = x$

Step 3

Divide each term in $4 \sqrt{4 - t^{2}} = x$ by $4$ and simplify.

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Step 3.1

Divide each term in $4 \sqrt{4 - t^{2}} = x$ by $4$ .

$\frac{4 \sqrt{4 - t^{2}}}{4} = \frac{x}{4}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Reduce the expression by cancelling the common factors.

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Step 3.2.1.1

Cancel the common factor of $4$ .

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Step 3.2.1.1.1

Cancel the common factor.

$\frac{4 \sqrt{4 - t^{2}}}{4} = \frac{x}{4}$

Step 3.2.1.1.2

Divide $\sqrt{4 - t^{2}}$ by $1$ .

$\sqrt{4 - t^{2}} = \frac{x}{4}$

Step 3.2.1.2

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

$\sqrt{2^{2} - t^{2}} = \frac{x}{4}$

Step 3.2.2

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

$\sqrt{(2 + t) (2 - t)} = \frac{x}{4}$

Step 4

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{(2 + t) (2 - t)}}^{2} = {(\frac{x}{4})}^{2}$

Step 5

Simplify each side of the equation.

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Step 5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(2 + t) (2 - t)}$ as ${((2 + t) (2 - t))}^{\frac{1}{2}}$ .

${({((2 + t) (2 - t))}^{\frac{1}{2}})}^{2} = {(\frac{x}{4})}^{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

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

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Step 5.2.1.1

Multiply the exponents in ${({((2 + t) (2 - t))}^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 5.2.1.1.2

Cancel the common factor of $2$ .

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Step 5.2.1.1.2.1

Cancel the common factor.

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

Step 5.2.1.1.2.2

Rewrite the expression.

${((2 + t) (2 - t))}^{1} = {(\frac{x}{4})}^{2}$

Step 5.2.1.2

Expand $(2 + t) (2 - t)$ using the FOIL Method.

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Step 5.2.1.2.1

Apply the distributive property.

${(2 (2 - t) + t (2 - t))}^{1} = {(\frac{x}{4})}^{2}$

Step 5.2.1.2.2

Apply the distributive property.

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

Step 5.2.1.2.3

Apply the distributive property.

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

Step 5.2.1.3

Simplify and combine like terms.

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Step 5.2.1.3.1

Simplify each term.

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Step 5.2.1.3.1.1

Multiply $2$ by $2$ .

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

Step 5.2.1.3.1.2

Multiply $- 1$ by $2$ .

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

Step 5.2.1.3.1.3

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

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

Step 5.2.1.3.1.4

Rewrite using the commutative property of multiplication.

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

Step 5.2.1.3.1.5

Multiply $t$ by $t$ by adding the exponents.

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Step 5.2.1.3.1.5.1

Move $t$ .

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

Step 5.2.1.3.1.5.2

Multiply $t$ by $t$ .

${(4 - 2 t + 2 t - t^{2})}^{1} = {(\frac{x}{4})}^{2}$

Step 5.2.1.3.2

Add $- 2 t$ and $2 t$ .

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

Step 5.2.1.3.3

Add $4$ and $0$ .

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

Step 5.2.1.4

Simplify.

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

Step 5.3

Simplify the right side.

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Step 5.3.1

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

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Step 5.3.1.1

Apply the product rule to $\frac{x}{4}$ .

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

Step 5.3.1.2

Raise $4$ to the power of $2$ .

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

Step 6

Solve for $t$ .

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Step 6.1

Subtract $4$ from both sides of the equation.

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

Step 6.2

Divide each term in $- t^{2} = \frac{x^{2}}{16} - 4$ by $- 1$ and simplify.

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Step 6.2.1

Divide each term in $- t^{2} = \frac{x^{2}}{16} - 4$ by $- 1$ .

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

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Dividing two negative values results in a positive value.

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

Step 6.2.2.2

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

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

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Simplify each term.

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Step 6.2.3.1.1

Move the negative one from the denominator of $\frac{\frac{x^{2}}{16}}{- 1}$ .

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

Step 6.2.3.1.2

Rewrite $- 1 \cdot \frac{x^{2}}{16}$ as $- \frac{x^{2}}{16}$ .

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

Step 6.2.3.1.3

Divide $- 4$ by $- 1$ .

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

Step 6.3

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

$t = \pm \sqrt{- \frac{x^{2}}{16} + 4}$

Step 6.4

Simplify $\pm \sqrt{- \frac{x^{2}}{16} + 4}$ .

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Step 6.4.1

Simplify the expression.

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Step 6.4.1.1

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

$t = \pm \sqrt{- \frac{x^{2}}{16} + 2^{2}}$

Step 6.4.1.2

Rewrite $\frac{x^{2}}{16}$ as ${(\frac{x}{4})}^{2}$ .

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

Step 6.4.1.3

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

$t = \pm \sqrt{2^{2} - {(\frac{x}{4})}^{2}}$

Step 6.4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2$ and $b = \frac{x}{4}$ .

$t = \pm \sqrt{(2 + \frac{x}{4}) (2 - \frac{x}{4})}$

Step 6.4.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$t = \pm \sqrt{(2 \cdot \frac{4}{4} + \frac{x}{4}) (2 - \frac{x}{4})}$

Step 6.4.4

Combine $2$ and $\frac{4}{4}$ .

$t = \pm \sqrt{(\frac{2 \cdot 4}{4} + \frac{x}{4}) (2 - \frac{x}{4})}$

Step 6.4.5

Combine the numerators over the common denominator.

$t = \pm \sqrt{\frac{2 \cdot 4 + x}{4} (2 - \frac{x}{4})}$

Step 6.4.6

Multiply $2$ by $4$ .

$t = \pm \sqrt{\frac{8 + x}{4} (2 - \frac{x}{4})}$

Step 6.4.7

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$t = \pm \sqrt{\frac{8 + x}{4} (2 \cdot \frac{4}{4} - \frac{x}{4})}$

Step 6.4.8

Combine $2$ and $\frac{4}{4}$ .

$t = \pm \sqrt{\frac{8 + x}{4} (\frac{2 \cdot 4}{4} - \frac{x}{4})}$

Step 6.4.9

Combine the numerators over the common denominator.

$t = \pm \sqrt{\frac{8 + x}{4} \cdot \frac{2 \cdot 4 - x}{4}}$

Step 6.4.10

Multiply $2$ by $4$ .

$t = \pm \sqrt{\frac{8 + x}{4} \cdot \frac{8 - x}{4}}$

Step 6.4.11

Multiply $\frac{8 + x}{4}$ by $\frac{8 - x}{4}$ .

$t = \pm \sqrt{\frac{(8 + x) (8 - x)}{4 \cdot 4}}$

Step 6.4.12

Multiply $4$ by $4$ .

$t = \pm \sqrt{\frac{(8 + x) (8 - x)}{16}}$

Step 6.4.13

Rewrite $\frac{(8 + x) (8 - x)}{16}$ as ${(\frac{1}{4})}^{2} ((8 + x) (8 - x))$ .

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Step 6.4.13.1

Factor the perfect power $1^{2}$ out of $(8 + x) (8 - x)$ .

$t = \pm \sqrt{\frac{1^{2} ((8 + x) (8 - x))}{16}}$

Step 6.4.13.2

Factor the perfect power $4^{2}$ out of $16$ .

$t = \pm \sqrt{\frac{1^{2} ((8 + x) (8 - x))}{4^{2} \cdot 1}}$

Step 6.4.13.3

Rearrange the fraction $\frac{1^{2} ((8 + x) (8 - x))}{4^{2} \cdot 1}$ .

$t = \pm \sqrt{{(\frac{1}{4})}^{2} ((8 + x) (8 - x))}$

Step 6.4.14

Pull terms out from under the radical.

$t = \pm \frac{1}{4} \sqrt{(8 + x) (8 - x)}$

Step 6.4.15

Combine $\frac{1}{4}$ and $\sqrt{(8 + x) (8 - x)}$ .

$t = \pm \frac{\sqrt{(8 + x) (8 - x)}}{4}$

Step 6.5

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

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Step 6.5.1

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

$t = \frac{\sqrt{(8 + x) (8 - x)}}{4}$

Step 6.5.2

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

$t = - \frac{\sqrt{(8 + x) (8 - x)}}{4}$

Step 6.5.3

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

$t = \frac{\sqrt{(8 + x) (8 - x)}}{4}$

$t = - \frac{\sqrt{(8 + x) (8 - x)}}{4}$

$t = \frac{\sqrt{(8 + x) (8 - x)}}{4}$

$t = - \frac{\sqrt{(8 + x) (8 - x)}}{4}$

$t = \frac{\sqrt{(8 + x) (8 - x)}}{4}$

$t = - \frac{\sqrt{(8 + x) (8 - x)}}{4}$

Step 7

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

$y = 3 (1 - \sqrt{4 - {(\frac{\sqrt{(8 + x) (8 - x)}}{4}, - \frac{\sqrt{(8 + x) (8 - x)}}{4})}^{2}})$

Step 8

Simplify $3 (1 - \sqrt{4 - {(\frac{\sqrt{(8 + x) (8 - x)}}{4}, - \frac{\sqrt{(8 + x) (8 - x)}}{4})}^{2}})$ .

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Step 8.1

Simplify each term.

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Step 8.1.1

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

$y = 3 (1 - \sqrt{2^{2} - {(\frac{\sqrt{(8 + x) (8 - x)}}{4}, - \frac{\sqrt{(8 + x) (8 - x)}}{4})}^{2}})$

Step 8.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2$ and $b = (\frac{\sqrt{(8 + x) (8 - x)}}{4}, - \frac{\sqrt{(8 + x) (8 - x)}}{4})$ .

$y = 3 (1 - \sqrt{(2 + (\frac{\sqrt{(8 + x) (8 - x)}}{4}, - \frac{\sqrt{(8 + x) (8 - x)}}{4})) (2 - (\frac{\sqrt{(8 + x) (8 - x)}}{4}, - \frac{\sqrt{(8 + x) (8 - x)}}{4}))})$

Step 8.1.3

Simplify.

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Step 8.1.3.1

Multiply $- 1$ by each element of the matrix.

$y = 3 (1 - \sqrt{(2 + (\frac{\sqrt{(8 + x) (8 - x)}}{4}, - \frac{\sqrt{(8 + x) (8 - x)}}{4})) (2 + (- \frac{\sqrt{(8 + x) (8 - x)}}{4}, - - \frac{\sqrt{(8 + x) (8 - x)}}{4}))})$

Step 8.1.3.2

Multiply $- - \frac{\sqrt{(8 + x) (8 - x)}}{4}$ .

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Step 8.1.3.2.1

Multiply $- 1$ by $- 1$ .

$y = 3 (1 - \sqrt{(2 + (\frac{\sqrt{(8 + x) (8 - x)}}{4}, - \frac{\sqrt{(8 + x) (8 - x)}}{4})) (2 + (- \frac{\sqrt{(8 + x) (8 - x)}}{4}, 1 (\frac{\sqrt{(8 + x) (8 - x)}}{4})))})$

Step 8.1.3.2.2

Multiply $\frac{\sqrt{(8 + x) (8 - x)}}{4}$ by $1$ .

$y = 3 (1 - \sqrt{(2 + (\frac{\sqrt{(8 + x) (8 - x)}}{4}, - \frac{\sqrt{(8 + x) (8 - x)}}{4})) (2 + (- \frac{\sqrt{(8 + x) (8 - x)}}{4}, \frac{\sqrt{(8 + x) (8 - x)}}{4}))})$

Step 8.2

Simplify by multiplying through.

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Step 8.2.1

Apply the distributive property.

$y = 3 \cdot 1 + 3 (- \sqrt{(2 + (\frac{\sqrt{(8 + x) (8 - x)}}{4}, - \frac{\sqrt{(8 + x) (8 - x)}}{4})) (2 + (- \frac{\sqrt{(8 + x) (8 - x)}}{4}, \frac{\sqrt{(8 + x) (8 - x)}}{4}))})$

Step 8.2.2

Simplify the expression.

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Step 8.2.2.1

Multiply $3$ by $1$ .

$y = 3 + 3 (- \sqrt{(2 + (\frac{\sqrt{(8 + x) (8 - x)}}{4}, - \frac{\sqrt{(8 + x) (8 - x)}}{4})) (2 + (- \frac{\sqrt{(8 + x) (8 - x)}}{4}, \frac{\sqrt{(8 + x) (8 - x)}}{4}))})$

Step 8.2.2.2

Multiply $- 1$ by $3$ .

$y = 3 - 3 \sqrt{(2 + (\frac{\sqrt{(8 + x) (8 - x)}}{4}, - \frac{\sqrt{(8 + x) (8 - x)}}{4})) (2 + (- \frac{\sqrt{(8 + x) (8 - x)}}{4}, \frac{\sqrt{(8 + x) (8 - x)}}{4}))}$

Step 8.2.2.3

Reorder $8$ and $x$ .

$y = 3 - 3 \sqrt{(2 + (\frac{\sqrt{(x + 8) (8 - x)}}{4}, - \frac{\sqrt{(8 + x) (8 - x)}}{4})) (2 + (- \frac{\sqrt{(8 + x) (8 - x)}}{4}, \frac{\sqrt{(8 + x) (8 - x)}}{4}))}$

Step 8.2.2.4

Reorder $8$ and $- x$ .

$y = 3 - 3 \sqrt{(2 + (\frac{\sqrt{(x + 8) (- x + 8)}}{4}, - \frac{\sqrt{(8 + x) (8 - x)}}{4})) (2 + (- \frac{\sqrt{(8 + x) (8 - x)}}{4}, \frac{\sqrt{(8 + x) (8 - x)}}{4}))}$

Step 8.2.2.5

Reorder $8$ and $x$ .

$y = 3 - 3 \sqrt{(2 + (\frac{\sqrt{(x + 8) (- x + 8)}}{4}, - \frac{\sqrt{(x + 8) (8 - x)}}{4})) (2 + (- \frac{\sqrt{(8 + x) (8 - x)}}{4}, \frac{\sqrt{(8 + x) (8 - x)}}{4}))}$

Step 8.2.2.6

Reorder $8$ and $- x$ .

$y = 3 - 3 \sqrt{(2 + (\frac{\sqrt{(x + 8) (- x + 8)}}{4}, - \frac{\sqrt{(x + 8) (- x + 8)}}{4})) (2 + (- \frac{\sqrt{(8 + x) (8 - x)}}{4}, \frac{\sqrt{(8 + x) (8 - x)}}{4}))}$

Step 8.2.2.7

Reorder $8$ and $x$ .

$y = 3 - 3 \sqrt{(2 + (\frac{\sqrt{(x + 8) (- x + 8)}}{4}, - \frac{\sqrt{(x + 8) (- x + 8)}}{4})) (2 + (- \frac{\sqrt{(x + 8) (8 - x)}}{4}, \frac{\sqrt{(8 + x) (8 - x)}}{4}))}$

Step 8.2.2.8

Reorder $8$ and $- x$ .

$y = 3 - 3 \sqrt{(2 + (\frac{\sqrt{(x + 8) (- x + 8)}}{4}, - \frac{\sqrt{(x + 8) (- x + 8)}}{4})) (2 + (- \frac{\sqrt{(x + 8) (- x + 8)}}{4}, \frac{\sqrt{(8 + x) (8 - x)}}{4}))}$

Step 8.2.2.9

Reorder $8$ and $x$ .

$y = 3 - 3 \sqrt{(2 + (\frac{\sqrt{(x + 8) (- x + 8)}}{4}, - \frac{\sqrt{(x + 8) (- x + 8)}}{4})) (2 + (- \frac{\sqrt{(x + 8) (- x + 8)}}{4}, \frac{\sqrt{(x + 8) (8 - x)}}{4}))}$

Step 8.2.2.10

Reorder $8$ and $- x$ .

$y = 3 - 3 \sqrt{(2 + (\frac{\sqrt{(x + 8) (- x + 8)}}{4}, - \frac{\sqrt{(x + 8) (- x + 8)}}{4})) (2 + (- \frac{\sqrt{(x + 8) (- x + 8)}}{4}, \frac{\sqrt{(x + 8) (- x + 8)}}{4}))}$