Pre-Algebra Examples

$\frac{9 x^{2} - 18 x - 7}{2 x - 4} \cdot \frac{x - 2}{9 x^{2} - 49}$

Step 1

Factor by grouping.

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Step 1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 9 \cdot - 7 = - 63$ and whose sum is $b = - 18$ .

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Step 1.1.1

Factor $- 18$ out of $- 18 x$ .

$\frac{9 x^{2} - 18 (x) - 7}{2 x - 4} \cdot \frac{x - 2}{9 x^{2} - 49}$

Step 1.1.2

Rewrite $- 18$ as $3$ plus $- 21$

$\frac{9 x^{2} + (3 - 21) x - 7}{2 x - 4} \cdot \frac{x - 2}{9 x^{2} - 49}$

Step 1.1.3

Apply the distributive property.

$\frac{9 x^{2} + 3 x - 21 x - 7}{2 x - 4} \cdot \frac{x - 2}{9 x^{2} - 49}$

Step 1.2

Factor out the greatest common factor from each group.

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Step 1.2.1

Group the first two terms and the last two terms.

$\frac{(9 x^{2} + 3 x) - 21 x - 7}{2 x - 4} \cdot \frac{x - 2}{9 x^{2} - 49}$

Step 1.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{3 x (3 x + 1) - 7 (3 x + 1)}{2 x - 4} \cdot \frac{x - 2}{9 x^{2} - 49}$

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $3 x + 1$ .

$\frac{(3 x + 1) (3 x - 7)}{2 x - 4} \cdot \frac{x - 2}{9 x^{2} - 49}$

Step 2

Factor $2$ out of $2 x - 4$ .

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Step 2.1

Factor $2$ out of $2 x$ .

$\frac{(3 x + 1) (3 x - 7)}{2 (x) - 4} \cdot \frac{x - 2}{9 x^{2} - 49}$

Step 2.2

Factor $2$ out of $- 4$ .

$\frac{(3 x + 1) (3 x - 7)}{2 x + 2 \cdot - 2} \cdot \frac{x - 2}{9 x^{2} - 49}$

Step 2.3

Factor $2$ out of $2 x + 2 \cdot - 2$ .

$\frac{(3 x + 1) (3 x - 7)}{2 (x - 2)} \cdot \frac{x - 2}{9 x^{2} - 49}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

$\frac{(3 x + 1) (3 x - 7)}{2 (x - 2)} \cdot \frac{x - 2}{{(3 x)}^{2} - 49}$

Step 3.2

Rewrite $49$ as $7^{2}$ .

$\frac{(3 x + 1) (3 x - 7)}{2 (x - 2)} \cdot \frac{x - 2}{{(3 x)}^{2} - 7^{2}}$

Step 3.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 x$ and $b = 7$ .

$\frac{(3 x + 1) (3 x - 7)}{2 (x - 2)} \cdot \frac{x - 2}{(3 x + 7) (3 x - 7)}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $3 x - 7$ .

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Step 4.1.1

Factor $3 x - 7$ out of $(3 x + 1) (3 x - 7)$ .

$\frac{(3 x - 7) (3 x + 1)}{2 (x - 2)} \cdot \frac{x - 2}{(3 x + 7) (3 x - 7)}$

Step 4.1.2

Factor $3 x - 7$ out of $(3 x + 7) (3 x - 7)$ .

$\frac{(3 x - 7) (3 x + 1)}{2 (x - 2)} \cdot \frac{x - 2}{(3 x - 7) (3 x + 7)}$

Step 4.1.3

Cancel the common factor.

$\frac{(3 x - 7) (3 x + 1)}{2 (x - 2)} \cdot \frac{x - 2}{(3 x - 7) (3 x + 7)}$

Step 4.1.4

Rewrite the expression.

$\frac{3 x + 1}{2 (x - 2)} \cdot \frac{x - 2}{3 x + 7}$

Step 4.2

Cancel the common factor of $x - 2$ .

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Step 4.2.1

Factor $x - 2$ out of $2 (x - 2)$ .

$\frac{3 x + 1}{(x - 2) \cdot 2} \cdot \frac{x - 2}{3 x + 7}$

Step 4.2.2

Cancel the common factor.

$\frac{3 x + 1}{(x - 2) \cdot 2} \cdot \frac{x - 2}{3 x + 7}$

Step 4.2.3

Rewrite the expression.

$\frac{3 x + 1}{2} \cdot \frac{1}{3 x + 7}$

Step 4.3

Multiply $\frac{3 x + 1}{2}$ by $\frac{1}{3 x + 7}$ .

$\frac{3 x + 1}{2 (3 x + 7)}$