Basic Math Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

Rewrite $4$ as $2^{2}$ .

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

Step 1.3

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 $b = 2$ .

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

Step 2

Factor by grouping.

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Step 2.1

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 $a \cdot c = 3 \cdot - 21 = - 63$ and whose sum is $b = - 2$ .

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Step 2.1.1

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

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

Step 2.1.2

Rewrite $- 2$ as $7$ plus $- 9$

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

Step 2.1.3

Apply the distributive property.

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

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

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

Step 2.2.2

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

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

Step 2.3

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

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

Step 3

Factor $3$ out of $3 x - 9$ .

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Step 3.1

Factor $3$ out of $3 x$ .

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

Step 3.2

Factor $3$ out of $- 9$ .

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

Step 3.3

Factor $3$ out of $3 x + 3 \cdot - 3$ .

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

Step 4

Cancel the common factor of $7 x - 2$ .

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Step 4.1

Factor $7 x - 2$ out of $(7 x + 2) (7 x - 2)$ .

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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

Step 5

Cancel the common factor of $x - 3$ .

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

Cancel the common factor.

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

Step 5.4

Rewrite the expression.

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

Step 6

Combine $\frac{7 x + 2}{3 x + 7}$ and $3$ .

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

Step 7

Move $3$ to the left of $7 x + 2$ .

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