Basic Math Examples

$\frac{1}{{(c + 2)}^{2}} + \frac{1}{{(c - 2)}^{2}}$

Step 1

To write $\frac{1}{{(c + 2)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{{(c - 2)}^{2}}{{(c - 2)}^{2}}$ .

$\frac{1}{{(c + 2)}^{2}} \cdot \frac{{(c - 2)}^{2}}{{(c - 2)}^{2}} + \frac{1}{{(c - 2)}^{2}}$

Step 2

To write $\frac{1}{{(c - 2)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{{(c + 2)}^{2}}{{(c + 2)}^{2}}$ .

$\frac{1}{{(c + 2)}^{2}} \cdot \frac{{(c - 2)}^{2}}{{(c - 2)}^{2}} + \frac{1}{{(c - 2)}^{2}} \cdot \frac{{(c + 2)}^{2}}{{(c + 2)}^{2}}$

Step 3

Write each expression with a common denominator of ${(c + 2)}^{2} {(c - 2)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{1}{{(c + 2)}^{2}}$ by $\frac{{(c - 2)}^{2}}{{(c - 2)}^{2}}$ .

$\frac{{(c - 2)}^{2}}{{(c + 2)}^{2} {(c - 2)}^{2}} + \frac{1}{{(c - 2)}^{2}} \cdot \frac{{(c + 2)}^{2}}{{(c + 2)}^{2}}$

Step 3.2

Multiply $\frac{1}{{(c - 2)}^{2}}$ by $\frac{{(c + 2)}^{2}}{{(c + 2)}^{2}}$ .

$\frac{{(c - 2)}^{2}}{{(c + 2)}^{2} {(c - 2)}^{2}} + \frac{{(c + 2)}^{2}}{{(c - 2)}^{2} {(c + 2)}^{2}}$

Step 3.3

Reorder the factors of ${(c - 2)}^{2} {(c + 2)}^{2}$ .

$\frac{{(c - 2)}^{2}}{{(c + 2)}^{2} {(c - 2)}^{2}} + \frac{{(c + 2)}^{2}}{{(c + 2)}^{2} {(c - 2)}^{2}}$

Step 4

Combine the numerators over the common denominator.

$\frac{{(c - 2)}^{2} + {(c + 2)}^{2}}{{(c + 2)}^{2} {(c - 2)}^{2}}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite ${(c - 2)}^{2}$ as $(c - 2) (c - 2)$ .

$\frac{(c - 2) (c - 2) + {(c + 2)}^{2}}{{(c + 2)}^{2} {(c - 2)}^{2}}$

Step 5.2

Expand $(c - 2) (c - 2)$ using the FOIL Method.

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Step 5.2.1

Apply the distributive property.

$\frac{c (c - 2) - 2 (c - 2) + {(c + 2)}^{2}}{{(c + 2)}^{2} {(c - 2)}^{2}}$

Step 5.2.2

Apply the distributive property.

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

Step 5.2.3

Apply the distributive property.

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

Step 5.3

Simplify and combine like terms.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Multiply $c$ by $c$ .

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

Step 5.3.1.2

Move $- 2$ to the left of $c$ .

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

Step 5.3.1.3

Multiply $- 2$ by $- 2$ .

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

Step 5.3.2

Subtract $2 c$ from $- 2 c$ .

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

Step 5.4

Rewrite ${(c + 2)}^{2}$ as $(c + 2) (c + 2)$ .

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

Step 5.5

Expand $(c + 2) (c + 2)$ using the FOIL Method.

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Step 5.5.1

Apply the distributive property.

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

Step 5.5.2

Apply the distributive property.

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

Step 5.5.3

Apply the distributive property.

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

Step 5.6

Simplify and combine like terms.

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Step 5.6.1

Simplify each term.

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Step 5.6.1.1

Multiply $c$ by $c$ .

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

Step 5.6.1.2

Move $2$ to the left of $c$ .

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

Step 5.6.1.3

Multiply $2$ by $2$ .

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

Step 5.6.2

Add $2 c$ and $2 c$ .

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

Step 5.7

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

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

Step 5.8

Add $- 4 c$ and $4 c$ .

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

Step 5.9

Add $2 c^{2}$ and $0$ .

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

Step 5.10

Add $4$ and $4$ .

$\frac{2 c^{2} + 8}{{(c + 2)}^{2} {(c - 2)}^{2}}$

Step 5.11

Factor $2$ out of $2 c^{2} + 8$ .

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Step 5.11.1

Factor $2$ out of $2 c^{2}$ .

$\frac{2 (c^{2}) + 8}{{(c + 2)}^{2} {(c - 2)}^{2}}$

Step 5.11.2

Factor $2$ out of $8$ .

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

Step 5.11.3

Factor $2$ out of $2 c^{2} + 2 \cdot 4$ .

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