Basic Math Examples

$c^{4} + 4 = (c^{2} - 2 c + 2) (c^{2} + 2 c + 2)$

Step 1

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

$(c^{2} - 2 c + 2) (c^{2} + 2 c + 2) = c^{4} + 4$

Step 2

Simplify $(c^{2} - 2 c + 2) (c^{2} + 2 c + 2)$ .

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Step 2.1

Rewrite.

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

Step 2.2

Simplify by adding zeros.

$(c^{2} - 2 c + 2) (c^{2} + 2 c + 2) = c^{4} + 4$

Step 2.3

Expand $(c^{2} - 2 c + 2) (c^{2} + 2 c + 2)$ by multiplying each term in the first expression by each term in the second expression.

$c^{2} c^{2} + c^{2} (2 c) + c^{2} \cdot 2 - 2 c \cdot c^{2} - 2 c (2 c) - 2 c \cdot 2 + 2 c^{2} + 2 (2 c) + 2 \cdot 2 = c^{4} + 4$

Step 2.4

Simplify terms.

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Step 2.4.1

Combine the opposite terms in $c^{2} c^{2} + c^{2} (2 c) + c^{2} \cdot 2 - 2 c \cdot c^{2} - 2 c (2 c) - 2 c \cdot 2 + 2 c^{2} + 2 (2 c) + 2 \cdot 2$ .

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Step 2.4.1.1

Reorder the factors in the terms $c^{2} (2 c)$ and $- 2 c \cdot c^{2}$ .

$c^{2} c^{2} + 2 c^{2} c + c^{2} \cdot 2 - 2 c^{2} c - 2 c (2 c) - 2 c \cdot 2 + 2 c^{2} + 2 (2 c) + 2 \cdot 2 = c^{4} + 4$

Step 2.4.1.2

Subtract $2 c^{2} c$ from $2 c^{2} c$ .

$c^{2} c^{2} + c^{2} \cdot 2 + 0 - 2 c (2 c) - 2 c \cdot 2 + 2 c^{2} + 2 (2 c) + 2 \cdot 2 = c^{4} + 4$

Step 2.4.1.3

Add $c^{2} c^{2} + c^{2} \cdot 2$ and $0$ .

$c^{2} c^{2} + c^{2} \cdot 2 - 2 c (2 c) - 2 c \cdot 2 + 2 c^{2} + 2 (2 c) + 2 \cdot 2 = c^{4} + 4$

Step 2.4.1.4

Reorder the factors in the terms $- 2 c \cdot 2$ and $2 (2 c)$ .

$c^{2} c^{2} + c^{2} \cdot 2 - 2 c (2 c) - 2 \cdot 2 c + 2 c^{2} + 2 \cdot 2 c + 2 \cdot 2 = c^{4} + 4$

Step 2.4.1.5

Add $- 2 \cdot 2 c$ and $2 \cdot 2 c$ .

$c^{2} c^{2} + c^{2} \cdot 2 - 2 c (2 c) + 2 c^{2} + 0 + 2 \cdot 2 = c^{4} + 4$

Step 2.4.1.6

Add $c^{2} c^{2} + c^{2} \cdot 2 - 2 c (2 c) + 2 c^{2}$ and $0$ .

$c^{2} c^{2} + c^{2} \cdot 2 - 2 c (2 c) + 2 c^{2} + 2 \cdot 2 = c^{4} + 4$

Step 2.4.2

Simplify each term.

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Step 2.4.2.1

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

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Step 2.4.2.1.1

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

$c^{2 + 2} + c^{2} \cdot 2 - 2 c (2 c) + 2 c^{2} + 2 \cdot 2 = c^{4} + 4$

Step 2.4.2.1.2

Add $2$ and $2$ .

$c^{4} + c^{2} \cdot 2 - 2 c (2 c) + 2 c^{2} + 2 \cdot 2 = c^{4} + 4$

Step 2.4.2.2

Move $2$ to the left of $c^{2}$ .

$c^{4} + 2 \cdot c^{2} - 2 c (2 c) + 2 c^{2} + 2 \cdot 2 = c^{4} + 4$

Step 2.4.2.3

Rewrite using the commutative property of multiplication.

$c^{4} + 2 c^{2} - 2 \cdot 2 c \cdot c + 2 c^{2} + 2 \cdot 2 = c^{4} + 4$

Step 2.4.2.4

Multiply $c$ by $c$ by adding the exponents.

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Step 2.4.2.4.1

Move $c$ .

$c^{4} + 2 c^{2} - 2 \cdot 2 (c \cdot c) + 2 c^{2} + 2 \cdot 2 = c^{4} + 4$

Step 2.4.2.4.2

Multiply $c$ by $c$ .

$c^{4} + 2 c^{2} - 2 \cdot 2 c^{2} + 2 c^{2} + 2 \cdot 2 = c^{4} + 4$

Step 2.4.2.5

Multiply $- 2$ by $2$ .

$c^{4} + 2 c^{2} - 4 c^{2} + 2 c^{2} + 2 \cdot 2 = c^{4} + 4$

Step 2.4.2.6

Multiply $2$ by $2$ .

$c^{4} + 2 c^{2} - 4 c^{2} + 2 c^{2} + 4 = c^{4} + 4$

Step 2.4.3

Simplify by adding terms.

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Step 2.4.3.1

Subtract $4 c^{2}$ from $2 c^{2}$ .

$c^{4} - 2 c^{2} + 2 c^{2} + 4 = c^{4} + 4$

Step 2.4.3.2

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

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Step 2.4.3.2.1

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

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

Step 2.4.3.2.2

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

$c^{4} + 4 = c^{4} + 4$

Step 3

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

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Step 3.1

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

$c^{4} + 4 - c^{4} = 4$

Step 3.2

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

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Step 3.2.1

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

$0 + 4 = 4$

Step 3.2.2

Add $0$ and $4$ .

$4 = 4$

Step 4

Since $4 = 4$ , the equation will always be true for any value of $c$ .

All real numbers

Step 5

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$