Algebra Examples

$\frac{3 k^{2} + 24 k}{6 k^{2} + 12 k} \div \frac{D}{2 k^{2} - 12 k - 32} = 1$

Step 1

Factor each term.

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Step 1.1

Factor $3 k$ out of $3 k^{2} + 24 k$ .

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Step 1.1.1

Factor $3 k$ out of $3 k^{2}$ .

$\frac{3 k (k) + 24 k}{6 k^{2} + 12 k} \div \frac{D}{2 k^{2} - 12 k - 32} = 1$

Step 1.1.2

Factor $3 k$ out of $24 k$ .

$\frac{3 k (k) + 3 k (8)}{6 k^{2} + 12 k} \div \frac{D}{2 k^{2} - 12 k - 32} = 1$

Step 1.1.3

Factor $3 k$ out of $3 k (k) + 3 k (8)$ .

$\frac{3 k (k + 8)}{6 k^{2} + 12 k} \div \frac{D}{2 k^{2} - 12 k - 32} = 1$

Step 1.2

Factor $6 k$ out of $6 k^{2} + 12 k$ .

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Step 1.2.1

Factor $6 k$ out of $6 k^{2}$ .

$\frac{3 k (k + 8)}{6 k (k) + 12 k} \div \frac{D}{2 k^{2} - 12 k - 32} = 1$

Step 1.2.2

Factor $6 k$ out of $12 k$ .

$\frac{3 k (k + 8)}{6 k (k) + 6 k (2)} \div \frac{D}{2 k^{2} - 12 k - 32} = 1$

Step 1.2.3

Factor $6 k$ out of $6 k (k) + 6 k (2)$ .

$\frac{3 k (k + 8)}{6 k (k + 2)} \div \frac{D}{2 k^{2} - 12 k - 32} = 1$

Step 1.3

Reduce the expression $\frac{3 k (k + 8)}{6 k (k + 2)}$ by cancelling the common factors.

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Step 1.3.1

Factor $3$ out of $3 k (k + 8)$ .

$\frac{3 (k (k + 8))}{6 k (k + 2)} \div \frac{D}{2 k^{2} - 12 k - 32} = 1$

Step 1.3.2

Factor $3$ out of $6 k (k + 2)$ .

$\frac{3 (k (k + 8))}{3 (2 k (k + 2))} \div \frac{D}{2 k^{2} - 12 k - 32} = 1$

Step 1.3.3

Cancel the common factor.

$\frac{3 (k (k + 8))}{3 (2 k (k + 2))} \div \frac{D}{2 k^{2} - 12 k - 32} = 1$

Step 1.3.4

Rewrite the expression.

$\frac{k (k + 8)}{2 k (k + 2)} \div \frac{D}{2 k^{2} - 12 k - 32} = 1$

Step 1.4

Factor $2$ out of $2 k^{2} - 12 k - 32$ .

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Step 1.4.1

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

$\frac{k (k + 8)}{2 k (k + 2)} \div \frac{D}{2 (k^{2}) - 12 k - 32} = 1$

Step 1.4.2

Factor $2$ out of $- 12 k$ .

$\frac{k (k + 8)}{2 k (k + 2)} \div \frac{D}{2 (k^{2}) + 2 (- 6 k) - 32} = 1$

Step 1.4.3

Factor $2$ out of $- 32$ .

$\frac{k (k + 8)}{2 k (k + 2)} \div \frac{D}{2 k^{2} + 2 (- 6 k) + 2 \cdot - 16} = 1$

Step 1.4.4

Factor $2$ out of $2 k^{2} + 2 (- 6 k)$ .

$\frac{k (k + 8)}{2 k (k + 2)} \div \frac{D}{2 (k^{2} - 6 k) + 2 \cdot - 16} = 1$

Step 1.4.5

Factor $2$ out of $2 (k^{2} - 6 k) + 2 \cdot - 16$ .

$\frac{k (k + 8)}{2 k (k + 2)} \div \frac{D}{2 (k^{2} - 6 k - 16)} = 1$

Step 1.5

Factor.

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Step 1.5.1

Factor $k^{2} - 6 k - 16$ using the AC method.

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Step 1.5.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 16$ and whose sum is $- 6$ .

$- 8, 2$

Step 1.5.1.2

Write the factored form using these integers.

$\frac{k (k + 8)}{2 k (k + 2)} \div \frac{D}{2 ((k - 8) (k + 2))} = 1$

Step 1.5.2

Remove unnecessary parentheses.

$\frac{k (k + 8)}{2 k (k + 2)} \div \frac{D}{2 (k - 8) (k + 2)} = 1$

Step 1.6

To divide by a fraction, multiply by its reciprocal.

$\frac{k (k + 8)}{2 k (k + 2)} \cdot \frac{2 (k - 8) (k + 2)}{D} = 1$

Step 1.7

Cancel the common factor of $2 (k + 2)$ .

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Step 1.7.1

Factor $2 (k + 2)$ out of $2 k (k + 2)$ .

$\frac{k (k + 8)}{2 (k + 2) (k)} \cdot \frac{2 (k - 8) (k + 2)}{D} = 1$

Step 1.7.2

Factor $2 (k + 2)$ out of $2 (k - 8) (k + 2)$ .

$\frac{k (k + 8)}{2 (k + 2) (k)} \cdot \frac{2 (k + 2) (k - 8)}{D} = 1$

Step 1.7.3

Cancel the common factor.

$\frac{k (k + 8)}{2 (k + 2) k} \cdot \frac{2 (k + 2) (k - 8)}{D} = 1$

Step 1.7.4

Rewrite the expression.

$\frac{k (k + 8)}{k} \cdot \frac{k - 8}{D} = 1$

Step 1.8

Multiply $\frac{k (k + 8)}{k}$ by $\frac{k - 8}{D}$ .

$\frac{k (k + 8) (k - 8)}{k D} = 1$

Step 1.9

Cancel the common factor of $k$ .

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Step 1.9.1

Cancel the common factor.

$\frac{k (k + 8) (k - 8)}{k D} = 1$

Step 1.9.2

Rewrite the expression.

$\frac{(k + 8) (k - 8)}{D} = 1$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$D, 1$

Step 2.2

The LCM of one and any expression is the expression.

$D$

Step 3

Multiply each term in $\frac{(k + 8) (k - 8)}{D} = 1$ by $D$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{(k + 8) (k - 8)}{D} = 1$ by $D$ .

$\frac{(k + 8) (k - 8)}{D} D = 1 D$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $D$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{(k + 8) (k - 8)}{D} D = 1 D$

Step 3.2.1.2

Rewrite the expression.

$(k + 8) (k - 8) = 1 D$

Step 3.2.2

Expand $(k + 8) (k - 8)$ using the FOIL Method.

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Step 3.2.2.1

Apply the distributive property.

$k (k - 8) + 8 (k - 8) = 1 D$

Step 3.2.2.2

Apply the distributive property.

$k \cdot k + k \cdot - 8 + 8 (k - 8) = 1 D$

Step 3.2.2.3

Apply the distributive property.

$k \cdot k + k \cdot - 8 + 8 k + 8 \cdot - 8 = 1 D$

Step 3.2.3

Simplify terms.

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Step 3.2.3.1

Combine the opposite terms in $k \cdot k + k \cdot - 8 + 8 k + 8 \cdot - 8$ .

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Step 3.2.3.1.1

Reorder the factors in the terms $k \cdot - 8$ and $8 k$ .

$k \cdot k - 8 k + 8 k + 8 \cdot - 8 = 1 D$

Step 3.2.3.1.2

Add $- 8 k$ and $8 k$ .

$k \cdot k + 0 + 8 \cdot - 8 = 1 D$

Step 3.2.3.1.3

Add $k \cdot k$ and $0$ .

$k \cdot k + 8 \cdot - 8 = 1 D$

Step 3.2.3.2

Simplify each term.

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Step 3.2.3.2.1

Multiply $k$ by $k$ .

$k^{2} + 8 \cdot - 8 = 1 D$

Step 3.2.3.2.2

Multiply $8$ by $- 8$ .

$k^{2} - 64 = 1 D$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply $D$ by $1$ .

$k^{2} - 64 = D$

Step 4

Rewrite the equation as $D = k^{2} - 64$ .

$D = k^{2} - 64$