Finite Math Examples

$\frac{\frac{21 x - 147}{7 x - 63}}{\frac{x^{2} - 49}{x^{2} - 3 x - 54}}$

Step 1

Cancel the common factor of $21 x - 147$ and $7 x - 63$ .

Tap for more steps...

Step 1.1

Factor $7$ out of $21 x$ .

$\frac{\frac{7 (3 x) - 147}{7 x - 63}}{\frac{x^{2} - 49}{x^{2} - 3 x - 54}}$

Step 1.2

Factor $7$ out of $- 147$ .

$\frac{\frac{7 (3 x) + 7 \cdot - 21}{7 x - 63}}{\frac{x^{2} - 49}{x^{2} - 3 x - 54}}$

Step 1.3

Factor $7$ out of $7 (3 x) + 7 (- 21)$ .

$\frac{\frac{7 (3 x - 21)}{7 x - 63}}{\frac{x^{2} - 49}{x^{2} - 3 x - 54}}$

Step 1.4

Cancel the common factors.

Tap for more steps...

Step 1.4.1

Factor $7$ out of $7 x$ .

$\frac{\frac{7 (3 x - 21)}{7 (x) - 63}}{\frac{x^{2} - 49}{x^{2} - 3 x - 54}}$

Step 1.4.2

Factor $7$ out of $- 63$ .

$\frac{\frac{7 (3 x - 21)}{7 x + 7 \cdot - 9}}{\frac{x^{2} - 49}{x^{2} - 3 x - 54}}$

Step 1.4.3

Factor $7$ out of $7 x + 7 \cdot - 9$ .

$\frac{\frac{7 (3 x - 21)}{7 (x - 9)}}{\frac{x^{2} - 49}{x^{2} - 3 x - 54}}$

Step 1.4.4

Cancel the common factor.

$\frac{\frac{7 (3 x - 21)}{7 (x - 9)}}{\frac{x^{2} - 49}{x^{2} - 3 x - 54}}$

Step 1.4.5

Rewrite the expression.

$\frac{\frac{3 x - 21}{x - 9}}{\frac{x^{2} - 49}{x^{2} - 3 x - 54}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{3 x - 21}{x - 9} \cdot \frac{x^{2} - 3 x - 54}{x^{2} - 49}$

Step 3

Factor $3$ out of $3 x - 21$ .

Tap for more steps...

Step 3.1

Factor $3$ out of $3 x$ .

$\frac{3 (x) - 21}{x - 9} \cdot \frac{x^{2} - 3 x - 54}{x^{2} - 49}$

Step 3.2

Factor $3$ out of $- 21$ .

$\frac{3 x + 3 \cdot - 7}{x - 9} \cdot \frac{x^{2} - 3 x - 54}{x^{2} - 49}$

Step 3.3

Factor $3$ out of $3 x + 3 \cdot - 7$ .

$\frac{3 (x - 7)}{x - 9} \cdot \frac{x^{2} - 3 x - 54}{x^{2} - 49}$

Step 4

Factor $x^{2} - 3 x - 54$ using the AC method.

Tap for more steps...

Step 4.1

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 $- 54$ and whose sum is $- 3$ .

$- 9, 6$

Step 4.2

Write the factored form using these integers.

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

Step 5

Simplify the denominator.

Tap for more steps...

Step 5.1

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

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

Step 5.2

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

$\frac{3 (x - 7)}{x - 9} \cdot \frac{(x - 9) (x + 6)}{(x + 7) (x - 7)}$

Step 6

Simplify terms.

Tap for more steps...

Step 6.1

Cancel the common factor of $x - 7$ .

Tap for more steps...

Step 6.1.1

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

$\frac{(x - 7) \cdot 3}{x - 9} \cdot \frac{(x - 9) (x + 6)}{(x + 7) (x - 7)}$

Step 6.1.2

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

$\frac{(x - 7) \cdot 3}{x - 9} \cdot \frac{(x - 9) (x + 6)}{(x - 7) (x + 7)}$

Step 6.1.3

Cancel the common factor.

$\frac{(x - 7) \cdot 3}{x - 9} \cdot \frac{(x - 9) (x + 6)}{(x - 7) (x + 7)}$

Step 6.1.4

Rewrite the expression.

$\frac{3}{x - 9} \cdot \frac{(x - 9) (x + 6)}{x + 7}$

Step 6.2

Cancel the common factor of $x - 9$ .

Tap for more steps...

Step 6.2.1

Cancel the common factor.

$\frac{3}{x - 9} \cdot \frac{(x - 9) (x + 6)}{x + 7}$

Step 6.2.2

Rewrite the expression.

$3 \frac{x + 6}{x + 7}$

Step 6.3

Combine $3$ and $\frac{x + 6}{x + 7}$ .

$\frac{3 (x + 6)}{x + 7}$