Pre-Algebra Examples

$\frac{14 x^{2} + x - 3}{4 x - 8} \cdot \frac{x - 2}{49 x^{2} - 9}$

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 $a \cdot c = 14 \cdot - 3 = - 42$ and whose sum is $b = 1$ .

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Step 1.1.1

Multiply by $1$ .

$\frac{14 x^{2} + 1 x - 3}{4 x - 8} \cdot \frac{x - 2}{49 x^{2} - 9}$

Step 1.1.2

Rewrite $1$ as $- 6$ plus $7$

$\frac{14 x^{2} + (- 6 + 7) x - 3}{4 x - 8} \cdot \frac{x - 2}{49 x^{2} - 9}$

Step 1.1.3

Apply the distributive property.

$\frac{14 x^{2} - 6 x + 7 x - 3}{4 x - 8} \cdot \frac{x - 2}{49 x^{2} - 9}$

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.

$\frac{(14 x^{2} - 6 x) + 7 x - 3}{4 x - 8} \cdot \frac{x - 2}{49 x^{2} - 9}$

Step 1.2.2

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

$\frac{2 x (7 x - 3) + 1 (7 x - 3)}{4 x - 8} \cdot \frac{x - 2}{49 x^{2} - 9}$

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $7 x - 3$ .

$\frac{(7 x - 3) (2 x + 1)}{4 x - 8} \cdot \frac{x - 2}{49 x^{2} - 9}$

Step 2

Factor $4$ out of $4 x - 8$ .

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Step 2.1

Factor $4$ out of $4 x$ .

$\frac{(7 x - 3) (2 x + 1)}{4 (x) - 8} \cdot \frac{x - 2}{49 x^{2} - 9}$

Step 2.2

Factor $4$ out of $- 8$ .

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

Step 2.3

Factor $4$ out of $4 x + 4 \cdot - 2$ .

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

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $49 x^{2}$ as ${(7 x)}^{2}$ .

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

Step 3.2

Rewrite $9$ as $3^{2}$ .

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

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 $a = 7 x$ and $b = 3$ .

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

Step 4

Simplify terms.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

Cancel the common factor.

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

Step 4.1.3

Rewrite the expression.

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

Step 4.2

Cancel the common factor of $x - 2$ .

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Step 4.2.1

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

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

$\frac{2 x + 1}{4} \cdot \frac{1}{7 x + 3}$

Step 4.3

Multiply $\frac{2 x + 1}{4}$ by $\frac{1}{7 x + 3}$ .

$\frac{2 x + 1}{4 (7 x + 3)}$