Calculus Examples

${(x^{2} + y^{2})}^{2} = 4 x^{2} y$ , $(- 1, 1)$

Step 1

Write $x^{4} + 2 x^{2} y^{2} + y^{4} - 4 x^{2} y = 0$ as a function.

$f (x) = x^{4} + 2 x^{2} y^{2} + y^{4} - 4 x^{2} y$

Step 2

Subtract $4 x^{2} y$ from both sides of the equation.

${(x^{2} + y^{2})}^{2} - 4 x^{2} y = 0$

Step 3

Simplify each term.

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Step 3.1

Rewrite ${(x^{2} + y^{2})}^{2}$ as $(x^{2} + y^{2}) (x^{2} + y^{2})$ .

$(x^{2} + y^{2}) (x^{2} + y^{2}) - 4 x^{2} y = 0$

Step 3.2

Expand $(x^{2} + y^{2}) (x^{2} + y^{2})$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

$x^{2} (x^{2} + y^{2}) + y^{2} (x^{2} + y^{2}) - 4 x^{2} y = 0$

Step 3.2.2

Apply the distributive property.

$x^{2} x^{2} + x^{2} y^{2} + y^{2} (x^{2} + y^{2}) - 4 x^{2} y = 0$

Step 3.2.3

Apply the distributive property.

$x^{2} x^{2} + x^{2} y^{2} + y^{2} x^{2} + y^{2} y^{2} - 4 x^{2} y = 0$

Step 3.3

Simplify and combine like terms.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

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

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Step 3.3.1.1.1

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

$x^{2 + 2} + x^{2} y^{2} + y^{2} x^{2} + y^{2} y^{2} - 4 x^{2} y = 0$

Step 3.3.1.1.2

Add $2$ and $2$ .

$x^{4} + x^{2} y^{2} + y^{2} x^{2} + y^{2} y^{2} - 4 x^{2} y = 0$

Step 3.3.1.2

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

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Step 3.3.1.2.1

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

$x^{4} + x^{2} y^{2} + y^{2} x^{2} + y^{2 + 2} - 4 x^{2} y = 0$

Step 3.3.1.2.2

Add $2$ and $2$ .

$x^{4} + x^{2} y^{2} + y^{2} x^{2} + y^{4} - 4 x^{2} y = 0$

Step 3.3.2

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

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Step 3.3.2.1

Reorder $y^{2}$ and $x^{2}$ .

$x^{4} + x^{2} y^{2} + x^{2} y^{2} + y^{4} - 4 x^{2} y = 0$

Step 3.3.2.2

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

$x^{4} + 2 x^{2} y^{2} + y^{4} - 4 x^{2} y = 0$

Step 4

Check if the given point is on the graph of the given function.

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Step 4.1

Evaluate $f (x) = x^{4} + 2 x^{2} y^{2} + y^{4} - 4 x^{2} y$ at $x = - 1$ .

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Step 4.1.1

Replace the variable $x$ with $- 1$ in the expression.

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

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Raise $- 1$ to the power of $4$ .

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

Step 4.1.2.1.2

Raise $- 1$ to the power of $2$ .

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

Step 4.1.2.1.3

Multiply $2$ by $1$ .

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

Step 4.1.2.1.4

Raise $- 1$ to the power of $2$ .

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

Step 4.1.2.1.5

Multiply $- 4$ by $1$ .

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

Step 4.1.2.2

The final answer is $1 + 2 y^{2} + y^{4} - 4 y$ .

$1 + 2 y^{2} + y^{4} - 4 y$

Step 4.2

Since $1 + 2 y^{2} + y^{4} - 4 y$ $\neq$ $1$ , the point is not on the graph.

The point is not on the graph

Step 5