Precalculus Examples

$2 x^{4} + 3 x^{2} y^{2} + y^{4} + y^{2}$ , $x^{2} + y^{2} = 1$

Step 1

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

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

Step 2

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

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

Step 3

Simplify $\pm \sqrt{1 - y^{2}}$ .

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Step 3.1

Rewrite $1$ as $1^{2}$ .

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

Step 3.2

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

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

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

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

Step 5

Replace the variable $x$ with $\sqrt{(1 + y) (1 - y)}$ in the expression.

$2 {(\sqrt{(1 + y) (1 - y)})}^{4} + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6

Simplify each term.

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Step 6.1

Rewrite ${\sqrt{(1 + y) (1 - y)}}^{4}$ as ${((1 + y) (1 - y))}^{2}$ .

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Step 6.1.1

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

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

Step 6.1.2

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

$2 {((1 + y) (1 - y))}^{\frac{1}{2} \cdot 4} + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.1.3

Combine $\frac{1}{2}$ and $4$ .

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

Step 6.1.4

Cancel the common factor of $4$ and $2$ .

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Step 6.1.4.1

Factor $2$ out of $4$ .

$2 {((1 + y) (1 - y))}^{\frac{2 \cdot 2}{2}} + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.1.4.2

Cancel the common factors.

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Step 6.1.4.2.1

Factor $2$ out of $2$ .

$2 {((1 + y) (1 - y))}^{\frac{2 \cdot 2}{2 (1)}} + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.1.4.2.2

Cancel the common factor.

$2 {((1 + y) (1 - y))}^{\frac{2 \cdot 2}{2 \cdot 1}} + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.1.4.2.3

Rewrite the expression.

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

Step 6.1.4.2.4

Divide $2$ by $1$ .

$2 {((1 + y) (1 - y))}^{2} + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.2

Expand $(1 + y) (1 - y)$ using the FOIL Method.

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Step 6.2.1

Apply the distributive property.

$2 {(1 (1 - y) + y (1 - y))}^{2} + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.2.2

Apply the distributive property.

$2 {(1 \cdot 1 + 1 (- y) + y (1 - y))}^{2} + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.2.3

Apply the distributive property.

$2 {(1 \cdot 1 + 1 (- y) + y \cdot 1 + y (- y))}^{2} + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.3

Simplify and combine like terms.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Multiply $1$ by $1$ .

$2 {(1 + 1 (- y) + y \cdot 1 + y (- y))}^{2} + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.3.1.2

Multiply $- y$ by $1$ .

$2 {(1 - y + y \cdot 1 + y (- y))}^{2} + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.3.1.3

Multiply $y$ by $1$ .

$2 {(1 - y + y + y (- y))}^{2} + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.3.1.4

Rewrite using the commutative property of multiplication.

$2 {(1 - y + y - y \cdot y)}^{2} + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.3.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 6.3.1.5.1

Move $y$ .

$2 {(1 - y + y - (y \cdot y))}^{2} + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.3.1.5.2

Multiply $y$ by $y$ .

$2 {(1 - y + y - y^{2})}^{2} + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.3.2

Add $- y$ and $y$ .

$2 {(1 + 0 - y^{2})}^{2} + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.3.3

Add $1$ and $0$ .

$2 {(1 - y^{2})}^{2} + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.4

Rewrite ${(1 - y^{2})}^{2}$ as $(1 - y^{2}) (1 - y^{2})$ .

$2 ((1 - y^{2}) (1 - y^{2})) + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.5

Expand $(1 - y^{2}) (1 - y^{2})$ using the FOIL Method.

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Step 6.5.1

Apply the distributive property.

$2 (1 (1 - y^{2}) - y^{2} (1 - y^{2})) + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.5.2

Apply the distributive property.

$2 (1 \cdot 1 + 1 (- y^{2}) - y^{2} (1 - y^{2})) + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.5.3

Apply the distributive property.

$2 (1 \cdot 1 + 1 (- y^{2}) - y^{2} \cdot 1 - y^{2} (- y^{2})) + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.6

Simplify and combine like terms.

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Step 6.6.1

Simplify each term.

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Step 6.6.1.1

Multiply $1$ by $1$ .

$2 (1 + 1 (- y^{2}) - y^{2} \cdot 1 - y^{2} (- y^{2})) + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.6.1.2

Multiply $- y^{2}$ by $1$ .

$2 (1 - y^{2} - y^{2} \cdot 1 - y^{2} (- y^{2})) + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.6.1.3

Multiply $- 1$ by $1$ .

$2 (1 - y^{2} - y^{2} - y^{2} (- y^{2})) + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.6.1.4

Rewrite using the commutative property of multiplication.

$2 (1 - y^{2} - y^{2} - 1 \cdot - 1 y^{2} y^{2}) + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.6.1.5

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

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Step 6.6.1.5.1

Move $y^{2}$ .

$2 (1 - y^{2} - y^{2} - 1 \cdot - 1 (y^{2} y^{2})) + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.6.1.5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 (1 - y^{2} - y^{2} - 1 \cdot - 1 y^{2 + 2}) + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.6.1.5.3

Add $2$ and $2$ .

$2 (1 - y^{2} - y^{2} - 1 \cdot - 1 y^{4}) + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.6.1.6

Multiply $- 1$ by $- 1$ .

$2 (1 - y^{2} - y^{2} + 1 y^{4}) + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.6.1.7

Multiply $y^{4}$ by $1$ .

$2 (1 - y^{2} - y^{2} + y^{4}) + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.6.2

Subtract $y^{2}$ from $- y^{2}$ .

$2 (1 - 2 y^{2} + y^{4}) + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.7

Apply the distributive property.

$2 \cdot 1 + 2 (- 2 y^{2}) + 2 y^{4} + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.8

Simplify.

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Step 6.8.1

Multiply $2$ by $1$ .

$2 + 2 (- 2 y^{2}) + 2 y^{4} + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.8.2

Multiply $- 2$ by $2$ .

$2 - 4 y^{2} + 2 y^{4} + 3 {(\sqrt{(1 + y) (1 - y)})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.9

Rewrite ${\sqrt{(1 + y) (1 - y)}}^{2}$ as $(1 + y) (1 - y)$ .

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Step 6.9.1

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

$2 - 4 y^{2} + 2 y^{4} + 3 {({((1 + y) (1 - y))}^{\frac{1}{2}})}^{2} y^{2} + y^{4} + y^{2}$

Step 6.9.2

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

$2 - 4 y^{2} + 2 y^{4} + 3 {((1 + y) (1 - y))}^{\frac{1}{2} \cdot 2} y^{2} + y^{4} + y^{2}$

Step 6.9.3

Combine $\frac{1}{2}$ and $2$ .

$2 - 4 y^{2} + 2 y^{4} + 3 {((1 + y) (1 - y))}^{\frac{2}{2}} y^{2} + y^{4} + y^{2}$

Step 6.9.4

Cancel the common factor of $2$ .

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Step 6.9.4.1

Cancel the common factor.

$2 - 4 y^{2} + 2 y^{4} + 3 {((1 + y) (1 - y))}^{\frac{2}{2}} y^{2} + y^{4} + y^{2}$

Step 6.9.4.2

Rewrite the expression.

$2 - 4 y^{2} + 2 y^{4} + 3 {((1 + y) (1 - y))}^{1} y^{2} + y^{4} + y^{2}$

Step 6.9.5

Simplify.

$2 - 4 y^{2} + 2 y^{4} + 3 ((1 + y) (1 - y)) y^{2} + y^{4} + y^{2}$

Step 6.10

Expand $(1 + y) (1 - y)$ using the FOIL Method.

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Step 6.10.1

Apply the distributive property.

$2 - 4 y^{2} + 2 y^{4} + 3 (1 (1 - y) + y (1 - y)) y^{2} + y^{4} + y^{2}$

Step 6.10.2

Apply the distributive property.

$2 - 4 y^{2} + 2 y^{4} + 3 (1 \cdot 1 + 1 (- y) + y (1 - y)) y^{2} + y^{4} + y^{2}$

Step 6.10.3

Apply the distributive property.

$2 - 4 y^{2} + 2 y^{4} + 3 (1 \cdot 1 + 1 (- y) + y \cdot 1 + y (- y)) y^{2} + y^{4} + y^{2}$

Step 6.11

Simplify and combine like terms.

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Step 6.11.1

Simplify each term.

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Step 6.11.1.1

Multiply $1$ by $1$ .

$2 - 4 y^{2} + 2 y^{4} + 3 (1 + 1 (- y) + y \cdot 1 + y (- y)) y^{2} + y^{4} + y^{2}$

Step 6.11.1.2

Multiply $- y$ by $1$ .

$2 - 4 y^{2} + 2 y^{4} + 3 (1 - y + y \cdot 1 + y (- y)) y^{2} + y^{4} + y^{2}$

Step 6.11.1.3

Multiply $y$ by $1$ .

$2 - 4 y^{2} + 2 y^{4} + 3 (1 - y + y + y (- y)) y^{2} + y^{4} + y^{2}$

Step 6.11.1.4

Rewrite using the commutative property of multiplication.

$2 - 4 y^{2} + 2 y^{4} + 3 (1 - y + y - y \cdot y) y^{2} + y^{4} + y^{2}$

Step 6.11.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 6.11.1.5.1

Move $y$ .

$2 - 4 y^{2} + 2 y^{4} + 3 (1 - y + y - (y \cdot y)) y^{2} + y^{4} + y^{2}$

Step 6.11.1.5.2

Multiply $y$ by $y$ .

$2 - 4 y^{2} + 2 y^{4} + 3 (1 - y + y - y^{2}) y^{2} + y^{4} + y^{2}$

Step 6.11.2

Add $- y$ and $y$ .

$2 - 4 y^{2} + 2 y^{4} + 3 (1 + 0 - y^{2}) y^{2} + y^{4} + y^{2}$

Step 6.11.3

Add $1$ and $0$ .

$2 - 4 y^{2} + 2 y^{4} + 3 (1 - y^{2}) y^{2} + y^{4} + y^{2}$

Step 6.12

Apply the distributive property.

$2 - 4 y^{2} + 2 y^{4} + (3 \cdot 1 + 3 (- y^{2})) y^{2} + y^{4} + y^{2}$

Step 6.13

Multiply $3$ by $1$ .

$2 - 4 y^{2} + 2 y^{4} + (3 + 3 (- y^{2})) y^{2} + y^{4} + y^{2}$

Step 6.14

Multiply $- 1$ by $3$ .

$2 - 4 y^{2} + 2 y^{4} + (3 - 3 y^{2}) y^{2} + y^{4} + y^{2}$

Step 6.15

Apply the distributive property.

$2 - 4 y^{2} + 2 y^{4} + 3 y^{2} - 3 y^{2} y^{2} + y^{4} + y^{2}$

Step 6.16

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

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Step 6.16.1

Move $y^{2}$ .

$2 - 4 y^{2} + 2 y^{4} + 3 y^{2} - 3 (y^{2} y^{2}) + y^{4} + y^{2}$

Step 6.16.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 - 4 y^{2} + 2 y^{4} + 3 y^{2} - 3 y^{2 + 2} + y^{4} + y^{2}$

Step 6.16.3

Add $2$ and $2$ .

$2 - 4 y^{2} + 2 y^{4} + 3 y^{2} - 3 y^{4} + y^{4} + y^{2}$

Step 7

Simplify by adding terms.

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Step 7.1

Add $- 4 y^{2}$ and $3 y^{2}$ .

$2 + 2 y^{4} - y^{2} - 3 y^{4} + y^{4} + y^{2}$

Step 7.2

Combine the opposite terms in $2 + 2 y^{4} - y^{2} - 3 y^{4} + y^{4} + y^{2}$ .

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Step 7.2.1

Add $- y^{2}$ and $y^{2}$ .

$2 + 2 y^{4} - 3 y^{4} + y^{4} + 0$

Step 7.2.2

Add $2 + 2 y^{4} - 3 y^{4} + y^{4}$ and $0$ .

$2 + 2 y^{4} - 3 y^{4} + y^{4}$

Step 7.3

Subtract $3 y^{4}$ from $2 y^{4}$ .

$2 - y^{4} + y^{4}$

Step 7.4

Combine the opposite terms in $2 - y^{4} + y^{4}$ .

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Step 7.4.1

Add $- y^{4}$ and $y^{4}$ .

$2 + 0$

Step 7.4.2

Add $2$ and $0$ .

$2$