Finite Math Examples

$\frac{\frac{2 x^{2} + 3 x - 2}{x^{2} - 14 x + 49}}{\frac{2 x^{2} + 2 x - 4}{42 - 6 x}}$

Step 1

Cancel the common factor of $2 x^{2} + 2 x - 4$ and $42 - 6 x$ .

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Step 1.1

Factor $2$ out of $2 x^{2}$ .

$\frac{\frac{2 x^{2} + 3 x - 2}{x^{2} - 14 x + 49}}{\frac{2 (x^{2}) + 2 x - 4}{42 - 6 x}}$

Step 1.2

Factor $2$ out of $2 x$ .

$\frac{\frac{2 x^{2} + 3 x - 2}{x^{2} - 14 x + 49}}{\frac{2 (x^{2}) + 2 (x) - 4}{42 - 6 x}}$

Step 1.3

Factor $2$ out of $2 (x^{2}) + 2 (x)$ .

$\frac{\frac{2 x^{2} + 3 x - 2}{x^{2} - 14 x + 49}}{\frac{2 (x^{2} + x) - 4}{42 - 6 x}}$

Step 1.4

Factor $2$ out of $- 4$ .

$\frac{\frac{2 x^{2} + 3 x - 2}{x^{2} - 14 x + 49}}{\frac{2 (x^{2} + x) + 2 (- 2)}{42 - 6 x}}$

Step 1.5

Factor $2$ out of $2 (x^{2} + x) + 2 (- 2)$ .

$\frac{\frac{2 x^{2} + 3 x - 2}{x^{2} - 14 x + 49}}{\frac{2 (x^{2} + x - 2)}{42 - 6 x}}$

Step 1.6

Cancel the common factors.

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Step 1.6.1

Factor $2$ out of $42$ .

$\frac{\frac{2 x^{2} + 3 x - 2}{x^{2} - 14 x + 49}}{\frac{2 (x^{2} + x - 2)}{2 (21) - 6 x}}$

Step 1.6.2

Factor $2$ out of $- 6 x$ .

$\frac{\frac{2 x^{2} + 3 x - 2}{x^{2} - 14 x + 49}}{\frac{2 (x^{2} + x - 2)}{2 (21) + 2 (- 3 x)}}$

Step 1.6.3

Factor $2$ out of $2 (21) + 2 (- 3 x)$ .

$\frac{\frac{2 x^{2} + 3 x - 2}{x^{2} - 14 x + 49}}{\frac{2 (x^{2} + x - 2)}{2 (21 - 3 x)}}$

Step 1.6.4

Cancel the common factor.

$\frac{\frac{2 x^{2} + 3 x - 2}{x^{2} - 14 x + 49}}{\frac{2 (x^{2} + x - 2)}{2 (21 - 3 x)}}$

Step 1.6.5

Rewrite the expression.

$\frac{\frac{2 x^{2} + 3 x - 2}{x^{2} - 14 x + 49}}{\frac{x^{2} + x - 2}{21 - 3 x}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 x^{2} + 3 x - 2}{x^{2} - 14 x + 49} \cdot \frac{21 - 3 x}{x^{2} + x - 2}$

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 $a \cdot c = 2 \cdot - 2 = - 4$ and whose sum is $b = 3$ .

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Step 3.1.1

Factor $3$ out of $3 x$ .

$\frac{2 x^{2} + 3 (x) - 2}{x^{2} - 14 x + 49} \cdot \frac{21 - 3 x}{x^{2} + x - 2}$

Step 3.1.2

Rewrite $3$ as $- 1$ plus $4$

$\frac{2 x^{2} + (- 1 + 4) x - 2}{x^{2} - 14 x + 49} \cdot \frac{21 - 3 x}{x^{2} + x - 2}$

Step 3.1.3

Apply the distributive property.

$\frac{2 x^{2} - 1 x + 4 x - 2}{x^{2} - 14 x + 49} \cdot \frac{21 - 3 x}{x^{2} + x - 2}$

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.

$\frac{(2 x^{2} - 1 x) + 4 x - 2}{x^{2} - 14 x + 49} \cdot \frac{21 - 3 x}{x^{2} + x - 2}$

Step 3.2.2

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

$\frac{x (2 x - 1) + 2 (2 x - 1)}{x^{2} - 14 x + 49} \cdot \frac{21 - 3 x}{x^{2} + x - 2}$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$\frac{(2 x - 1) (x + 2)}{x^{2} - 14 x + 49} \cdot \frac{21 - 3 x}{x^{2} + x - 2}$

Step 4

Factor using the perfect square rule.

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Step 4.1

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

$\frac{(2 x - 1) (x + 2)}{x^{2} - 14 x + 7^{2}} \cdot \frac{21 - 3 x}{x^{2} + x - 2}$

Step 4.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$14 x = 2 \cdot x \cdot 7$

Step 4.3

Rewrite the polynomial.

$\frac{(2 x - 1) (x + 2)}{x^{2} - 2 \cdot x \cdot 7 + 7^{2}} \cdot \frac{21 - 3 x}{x^{2} + x - 2}$

Step 4.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 7$ .

$\frac{(2 x - 1) (x + 2)}{{(x - 7)}^{2}} \cdot \frac{21 - 3 x}{x^{2} + x - 2}$

Step 5

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

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Step 5.1

Factor $3$ out of $21$ .

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

Step 5.2

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

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

Step 5.3

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

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

Step 6

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

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Step 6.1

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 $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 6.2

Write the factored form using these integers.

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

Step 7

Simplify terms.

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Step 7.1

Combine.

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

Step 7.2

Cancel the common factor of $x + 2$ .

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Step 7.2.1

Cancel the common factor.

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

Step 7.2.2

Rewrite the expression.

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

Step 7.3

Cancel the common factor of $7 - x$ and ${(x - 7)}^{2}$ .

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Step 7.3.1

Rewrite $7$ as $- 1 (- 7)$ .

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

Step 7.3.2

Factor $- 1$ out of $- x$ .

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

Step 7.3.3

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

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

Step 7.3.4

Reorder terms.

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

Step 7.3.5

Factor $x - 7$ out of $(2 x - 1) \cdot 3 (- 1 (x - 7))$ .

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

Step 7.3.6

Cancel the common factors.

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Step 7.3.6.1

Factor $x - 7$ out of ${(x - 7)}^{2} (x - 1)$ .

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

Step 7.3.6.2

Cancel the common factor.

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

Step 7.3.6.3

Rewrite the expression.

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

Step 7.4

Simplify the expression.

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Step 7.4.1

Multiply $- 1$ by $3$ .

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

Step 7.4.2

Move $- 3$ to the left of $2 x - 1$ .

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

Step 7.4.3

Move the negative in front of the fraction.

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