Calculus Examples

$2 {(x^{2} + y^{2})}^{2} = 25 (x^{2} - y^{2})$

Step 1

Subtract $25 (x^{2} - y^{2})$ from both sides of the equation.

$2 {(x^{2} + y^{2})}^{2} - 25 (x^{2} - y^{2}) = 0$

Step 2

Simplify each term.

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Step 2.1

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

$2 ((x^{2} + y^{2}) (x^{2} + y^{2})) - 25 (x^{2} - y^{2}) = 0$

Step 2.2

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

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Step 2.2.1

Apply the distributive property.

$2 (x^{2} (x^{2} + y^{2}) + y^{2} (x^{2} + y^{2})) - 25 (x^{2} - y^{2}) = 0$

Step 2.2.2

Apply the distributive property.

$2 (x^{2} x^{2} + x^{2} y^{2} + y^{2} (x^{2} + y^{2})) - 25 (x^{2} - y^{2}) = 0$

Step 2.2.3

Apply the distributive property.

$2 (x^{2} x^{2} + x^{2} y^{2} + y^{2} x^{2} + y^{2} y^{2}) - 25 (x^{2} - y^{2}) = 0$

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

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

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Step 2.3.1.1.1

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

$2 (x^{2 + 2} + x^{2} y^{2} + y^{2} x^{2} + y^{2} y^{2}) - 25 (x^{2} - y^{2}) = 0$

Step 2.3.1.1.2

Add $2$ and $2$ .

$2 (x^{4} + x^{2} y^{2} + y^{2} x^{2} + y^{2} y^{2}) - 25 (x^{2} - y^{2}) = 0$

Step 2.3.1.2

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

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Step 2.3.1.2.1

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

$2 (x^{4} + x^{2} y^{2} + y^{2} x^{2} + y^{2 + 2}) - 25 (x^{2} - y^{2}) = 0$

Step 2.3.1.2.2

Add $2$ and $2$ .

$2 (x^{4} + x^{2} y^{2} + y^{2} x^{2} + y^{4}) - 25 (x^{2} - y^{2}) = 0$

Step 2.3.2

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

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Step 2.3.2.1

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

$2 (x^{4} + x^{2} y^{2} + x^{2} y^{2} + y^{4}) - 25 (x^{2} - y^{2}) = 0$

Step 2.3.2.2

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

$2 (x^{4} + 2 x^{2} y^{2} + y^{4}) - 25 (x^{2} - y^{2}) = 0$

Step 2.4

Apply the distributive property.

$2 x^{4} + 2 (2 x^{2} y^{2}) + 2 y^{4} - 25 (x^{2} - y^{2}) = 0$

Step 2.5

Multiply $2$ by $2$ .

$2 x^{4} + 4 (x^{2} y^{2}) + 2 y^{4} - 25 (x^{2} - y^{2}) = 0$

Step 2.6

Apply the distributive property.

$2 x^{4} + 4 x^{2} y^{2} + 2 y^{4} - 25 x^{2} - 25 (- y^{2}) = 0$

Step 2.7

Multiply $- 1$ by $- 25$ .

$2 x^{4} + 4 x^{2} y^{2} + 2 y^{4} - 25 x^{2} + 25 y^{2} = 0$

Step 3

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$a x^{2} + b x + c = 0$

Step 4

The standard form is $2 x^{4} + 4 x^{2} y^{2} + 2 y^{4} - 25 x^{2} + 25 y^{2}$ .

$2 x^{4} + 4 x^{2} y^{2} + 2 y^{4} - 25 x^{2} + 25 y^{2} = 0$

Step 5