Basic Math Examples

$\frac{75 n^{2} - 147}{5 n^{2} - 12 n + 7} \div \frac{35 n^{2} + 34 n - 21}{n^{2} - 8 n + 7}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{75 n^{2} - 147}{5 n^{2} - 12 n + 7} \cdot \frac{n^{2} - 8 n + 7}{35 n^{2} + 34 n - 21}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $3$ out of $75 n^{2} - 147$ .

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Step 2.1.1

Factor $3$ out of $75 n^{2}$ .

$\frac{3 (25 n^{2}) - 147}{5 n^{2} - 12 n + 7} \cdot \frac{n^{2} - 8 n + 7}{35 n^{2} + 34 n - 21}$

Step 2.1.2

Factor $3$ out of $- 147$ .

$\frac{3 (25 n^{2}) + 3 (- 49)}{5 n^{2} - 12 n + 7} \cdot \frac{n^{2} - 8 n + 7}{35 n^{2} + 34 n - 21}$

Step 2.1.3

Factor $3$ out of $3 (25 n^{2}) + 3 (- 49)$ .

$\frac{3 (25 n^{2} - 49)}{5 n^{2} - 12 n + 7} \cdot \frac{n^{2} - 8 n + 7}{35 n^{2} + 34 n - 21}$

Step 2.2

Rewrite $25 n^{2}$ as ${(5 n)}^{2}$ .

$\frac{3 ({(5 n)}^{2} - 49)}{5 n^{2} - 12 n + 7} \cdot \frac{n^{2} - 8 n + 7}{35 n^{2} + 34 n - 21}$

Step 2.3

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

$\frac{3 ({(5 n)}^{2} - 7^{2})}{5 n^{2} - 12 n + 7} \cdot \frac{n^{2} - 8 n + 7}{35 n^{2} + 34 n - 21}$

Step 2.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5 n$ and $b = 7$ .

$\frac{3 (5 n + 7) (5 n - 7)}{5 n^{2} - 12 n + 7} \cdot \frac{n^{2} - 8 n + 7}{35 n^{2} + 34 n - 21}$

Step 3

Factor by grouping.

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Step 3.1

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 $a \cdot c = 5 \cdot 7 = 35$ and whose sum is $b = - 12$ .

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Step 3.1.1

Factor $- 12$ out of $- 12 n$ .

$\frac{3 (5 n + 7) (5 n - 7)}{5 n^{2} - 12 (n) + 7} \cdot \frac{n^{2} - 8 n + 7}{35 n^{2} + 34 n - 21}$

Step 3.1.2

Rewrite $- 12$ as $- 5$ plus $- 7$

$\frac{3 (5 n + 7) (5 n - 7)}{5 n^{2} + (- 5 - 7) n + 7} \cdot \frac{n^{2} - 8 n + 7}{35 n^{2} + 34 n - 21}$

Step 3.1.3

Apply the distributive property.

$\frac{3 (5 n + 7) (5 n - 7)}{5 n^{2} - 5 n - 7 n + 7} \cdot \frac{n^{2} - 8 n + 7}{35 n^{2} + 34 n - 21}$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$\frac{3 (5 n + 7) (5 n - 7)}{(5 n^{2} - 5 n) - 7 n + 7} \cdot \frac{n^{2} - 8 n + 7}{35 n^{2} + 34 n - 21}$

Step 3.2.2

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

$\frac{3 (5 n + 7) (5 n - 7)}{5 n (n - 1) - 7 (n - 1)} \cdot \frac{n^{2} - 8 n + 7}{35 n^{2} + 34 n - 21}$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $n - 1$ .

$\frac{3 (5 n + 7) (5 n - 7)}{(n - 1) (5 n - 7)} \cdot \frac{n^{2} - 8 n + 7}{35 n^{2} + 34 n - 21}$

Step 4

Cancel the common factor of $5 n - 7$ .

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Step 4.1

Cancel the common factor.

$\frac{3 (5 n + 7) (5 n - 7)}{(n - 1) (5 n - 7)} \cdot \frac{n^{2} - 8 n + 7}{35 n^{2} + 34 n - 21}$

Step 4.2

Rewrite the expression.

$\frac{3 (5 n + 7)}{n - 1} \cdot \frac{n^{2} - 8 n + 7}{35 n^{2} + 34 n - 21}$

Step 5

Factor $n^{2} - 8 n + 7$ using the AC method.

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Step 5.1

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 $7$ and whose sum is $- 8$ .

$- 7, - 1$

Step 5.2

Write the factored form using these integers.

$\frac{3 (5 n + 7)}{n - 1} \cdot \frac{(n - 7) (n - 1)}{35 n^{2} + 34 n - 21}$

Step 6

Factor by grouping.

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Step 6.1

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 $a \cdot c = 35 \cdot - 21 = - 735$ and whose sum is $b = 34$ .

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Step 6.1.1

Factor $34$ out of $34 n$ .

$\frac{3 (5 n + 7)}{n - 1} \cdot \frac{(n - 7) (n - 1)}{35 n^{2} + 34 (n) - 21}$

Step 6.1.2

Rewrite $34$ as $- 15$ plus $49$

$\frac{3 (5 n + 7)}{n - 1} \cdot \frac{(n - 7) (n - 1)}{35 n^{2} + (- 15 + 49) n - 21}$

Step 6.1.3

Apply the distributive property.

$\frac{3 (5 n + 7)}{n - 1} \cdot \frac{(n - 7) (n - 1)}{35 n^{2} - 15 n + 49 n - 21}$

Step 6.2

Factor out the greatest common factor from each group.

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Step 6.2.1

Group the first two terms and the last two terms.

$\frac{3 (5 n + 7)}{n - 1} \cdot \frac{(n - 7) (n - 1)}{(35 n^{2} - 15 n) + 49 n - 21}$

Step 6.2.2

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

$\frac{3 (5 n + 7)}{n - 1} \cdot \frac{(n - 7) (n - 1)}{5 n (7 n - 3) + 7 (7 n - 3)}$

Step 6.3

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

$\frac{3 (5 n + 7)}{n - 1} \cdot \frac{(n - 7) (n - 1)}{(7 n - 3) (5 n + 7)}$

Step 7

Cancel the common factor of $5 n + 7$ .

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Step 7.1

Factor $5 n + 7$ out of $3 (5 n + 7)$ .

$\frac{(5 n + 7) \cdot 3}{n - 1} \cdot \frac{(n - 7) (n - 1)}{(7 n - 3) (5 n + 7)}$

Step 7.2

Factor $5 n + 7$ out of $(7 n - 3) (5 n + 7)$ .

$\frac{(5 n + 7) \cdot 3}{n - 1} \cdot \frac{(n - 7) (n - 1)}{(5 n + 7) (7 n - 3)}$

Step 7.3

Cancel the common factor.

$\frac{(5 n + 7) \cdot 3}{n - 1} \cdot \frac{(n - 7) (n - 1)}{(5 n + 7) (7 n - 3)}$

Step 7.4

Rewrite the expression.

$\frac{3}{n - 1} \cdot \frac{(n - 7) (n - 1)}{7 n - 3}$

Step 8

Cancel the common factor of $n - 1$ .

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Step 8.1

Factor $n - 1$ out of $(n - 7) (n - 1)$ .

$\frac{3}{n - 1} \cdot \frac{(n - 1) (n - 7)}{7 n - 3}$

Step 8.2

Cancel the common factor.

$\frac{3}{n - 1} \cdot \frac{(n - 1) (n - 7)}{7 n - 3}$

Step 8.3

Rewrite the expression.

$3 \frac{n - 7}{7 n - 3}$

Step 9

Combine $3$ and $\frac{n - 7}{7 n - 3}$ .

$\frac{3 (n - 7)}{7 n - 3}$