Algebra Examples

${(x^{2} + 3)}^{2} - {(x^{2} - 1)}^{2}$

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(x^{2} + 3)}^{2}$ as $(x^{2} + 3) (x^{2} + 3)$ .

$(x^{2} + 3) (x^{2} + 3) - {(x^{2} - 1)}^{2}$

Step 1.2

Expand $(x^{2} + 3) (x^{2} + 3)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$x^{2} (x^{2} + 3) + 3 (x^{2} + 3) - {(x^{2} - 1)}^{2}$

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

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

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Step 1.3.1.1.1

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

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

Step 1.3.1.1.2

Add $2$ and $2$ .

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

Step 1.3.1.2

Move $3$ to the left of $x^{2}$ .

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

Step 1.3.1.3

Multiply $3$ by $3$ .

$x^{4} + 3 x^{2} + 3 x^{2} + 9 - {(x^{2} - 1)}^{2}$

Step 1.3.2

Add $3 x^{2}$ and $3 x^{2}$ .

$x^{4} + 6 x^{2} + 9 - {(x^{2} - 1)}^{2}$

Step 1.4

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

$x^{4} + 6 x^{2} + 9 - ((x^{2} - 1) (x^{2} - 1))$

Step 1.5

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

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Step 1.5.1

Apply the distributive property.

$x^{4} + 6 x^{2} + 9 - (x^{2} (x^{2} - 1) - 1 (x^{2} - 1))$

Step 1.5.2

Apply the distributive property.

$x^{4} + 6 x^{2} + 9 - (x^{2} x^{2} + x^{2} \cdot - 1 - 1 (x^{2} - 1))$

Step 1.5.3

Apply the distributive property.

$x^{4} + 6 x^{2} + 9 - (x^{2} x^{2} + x^{2} \cdot - 1 - 1 x^{2} - 1 \cdot - 1)$

Step 1.6

Simplify and combine like terms.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

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

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Step 1.6.1.1.1

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

$x^{4} + 6 x^{2} + 9 - (x^{2 + 2} + x^{2} \cdot - 1 - 1 x^{2} - 1 \cdot - 1)$

Step 1.6.1.1.2

Add $2$ and $2$ .

$x^{4} + 6 x^{2} + 9 - (x^{4} + x^{2} \cdot - 1 - 1 x^{2} - 1 \cdot - 1)$

Step 1.6.1.2

Move $- 1$ to the left of $x^{2}$ .

$x^{4} + 6 x^{2} + 9 - (x^{4} - 1 \cdot x^{2} - 1 x^{2} - 1 \cdot - 1)$

Step 1.6.1.3

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

$x^{4} + 6 x^{2} + 9 - (x^{4} - x^{2} - 1 x^{2} - 1 \cdot - 1)$

Step 1.6.1.4

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

$x^{4} + 6 x^{2} + 9 - (x^{4} - x^{2} - x^{2} - 1 \cdot - 1)$

Step 1.6.1.5

Multiply $- 1$ by $- 1$ .

$x^{4} + 6 x^{2} + 9 - (x^{4} - x^{2} - x^{2} + 1)$

Step 1.6.2

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

$x^{4} + 6 x^{2} + 9 - (x^{4} - 2 x^{2} + 1)$

Step 1.7

Apply the distributive property.

$x^{4} + 6 x^{2} + 9 - x^{4} - (- 2 x^{2}) - 1 \cdot 1$

Step 1.8

Simplify.

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Step 1.8.1

Multiply $- 2$ by $- 1$ .

$x^{4} + 6 x^{2} + 9 - x^{4} + 2 x^{2} - 1 \cdot 1$

Step 1.8.2

Multiply $- 1$ by $1$ .

$x^{4} + 6 x^{2} + 9 - x^{4} + 2 x^{2} - 1$

Step 2

Simplify by adding terms.

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Step 2.1

Combine the opposite terms in $x^{4} + 6 x^{2} + 9 - x^{4} + 2 x^{2} - 1$ .

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Step 2.1.1

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

$6 x^{2} + 9 + 0 + 2 x^{2} - 1$

Step 2.1.2

Add $6 x^{2} + 9$ and $0$ .

$6 x^{2} + 9 + 2 x^{2} - 1$

Step 2.2

Add $6 x^{2}$ and $2 x^{2}$ .

$8 x^{2} + 9 - 1$

Step 2.3

Subtract $1$ from $9$ .

$8 x^{2} + 8$