Algebra Examples

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

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 2

Simplify each term.

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Step 2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1.1

Apply the product rule to $2 x y$ .

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

Step 2.1.2

Apply the product rule to $2 x$ .

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

Step 2.2

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

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

Step 3

Simplify ${(x^{2} + y^{2})}^{2}$ .

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Step 3.1

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

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

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.

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

Step 3.2.2

Apply the distributive property.

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

Step 3.2.3

Apply the distributive property.

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

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.

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

Step 3.3.1.1.2

Add $2$ and $2$ .

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

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.

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

Step 3.3.1.2.2

Add $2$ and $2$ .

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

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 $x^{2}$ .

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

Step 3.3.2.2

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

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

Step 4

Move all terms containing $x$ to the left side of the equation.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Simplify each term.

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Step 4.3.1

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

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

Step 4.3.2

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

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Step 4.3.2.1

Apply the distributive property.

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

Step 4.3.2.2

Apply the distributive property.

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

Step 4.3.2.3

Apply the distributive property.

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

Step 4.3.3

Simplify and combine like terms.

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Step 4.3.3.1

Simplify each term.

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Step 4.3.3.1.1

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

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Step 4.3.3.1.1.1

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

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

Step 4.3.3.1.1.2

Add $2$ and $2$ .

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

Step 4.3.3.1.2

Rewrite using the commutative property of multiplication.

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

Step 4.3.3.1.3

Rewrite using the commutative property of multiplication.

$x^{4} - x^{2} y^{2} - y^{2} x^{2} - 1 \cdot - 1 y^{2} y^{2} + 4 x^{2} y^{2} - x^{4} - 2 x^{2} y^{2} = y^{4}$

Step 4.3.3.1.4

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

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Step 4.3.3.1.4.1

Move $y^{2}$ .

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

Step 4.3.3.1.4.2

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

$x^{4} - x^{2} y^{2} - y^{2} x^{2} - 1 \cdot - 1 y^{2 + 2} + 4 x^{2} y^{2} - x^{4} - 2 x^{2} y^{2} = y^{4}$

Step 4.3.3.1.4.3

Add $2$ and $2$ .

$x^{4} - x^{2} y^{2} - y^{2} x^{2} - 1 \cdot - 1 y^{4} + 4 x^{2} y^{2} - x^{4} - 2 x^{2} y^{2} = y^{4}$

Step 4.3.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 4.3.3.1.6

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

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

Step 4.3.3.2

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

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Step 4.3.3.2.1

Move $y^{2}$ .

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

Step 4.3.3.2.2

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

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

Step 4.4

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

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Step 4.4.1

Subtract $x^{4}$ from $x^{4}$ .

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

Step 4.4.2

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

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

Step 4.5

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

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

Step 4.6

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

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Step 4.6.1

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

$y^{4} + 0 = y^{4}$

Step 4.6.2

Add $y^{4}$ and $0$ .

$y^{4} = y^{4}$

Step 5

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$| y | = | y |$

Step 6

Rewrite the absolute value equation as four equations without absolute value bars.

$y = y$

$y = - y$

$- y = y$

$- y = - y$

Step 7

After simplifying, there are only two unique equations to be solved.

$y = y$

$y = - y$

Step 8

Solve $y = y$ for $y$ .

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Step 8.1

Move all terms containing $y$ to the left side of the equation.

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Step 8.1.1

Subtract $y$ from both sides of the equation.

$y - y = 0$

Step 8.1.2

Subtract $y$ from $y$ .

$0 = 0$

Step 8.2

Since $0 = 0$ , the equation will always be true.

Always true

Step 9

Solve $y = - y$ for $y$ .

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Step 9.1

Move all terms containing $y$ to the left side of the equation.

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Step 9.1.1

Add $y$ to both sides of the equation.

$y + y = 0$

Step 9.1.2

Add $y$ and $y$ .

$2 y = 0$

Step 9.2

Divide each term in $2 y = 0$ by $2$ and simplify.

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Step 9.2.1

Divide each term in $2 y = 0$ by $2$ .

$\frac{2 y}{2} = \frac{0}{2}$

Step 9.2.2

Simplify the left side.

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Step 9.2.2.1

Cancel the common factor of $2$ .

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Step 9.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{0}{2}$

Step 9.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{0}{2}$

Step 9.2.3

Simplify the right side.

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Step 9.2.3.1

Divide $0$ by $2$ .

$y = 0$

Step 10

List all of the solutions.

$y = 0$

Step 11

The variable $x$ got canceled.

All real numbers

Step 12

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$