Algebra Examples

$\frac{x^{2} + x - 6}{x^{2} - 6 x + 5} \div \frac{x^{2} + 2 x - 3}{x^{2} - 7 x + 10}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} + x - 6}{x^{2} - 6 x + 5} \cdot \frac{x^{2} - 7 x + 10}{x^{2} + 2 x - 3}$

Step 2

Factor $x^{2} + x - 6$ using the AC method.

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Step 2.1

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 $- 6$ and whose sum is $1$ .

$- 2, 3$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 2) (x + 3)}{x^{2} - 6 x + 5} \cdot \frac{x^{2} - 7 x + 10}{x^{2} + 2 x - 3}$

Step 3

Factor $x^{2} - 6 x + 5$ using the AC method.

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Step 3.1

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 $5$ and whose sum is $- 6$ .

$- 5, - 1$

Step 3.2

Write the factored form using these integers.

$\frac{(x - 2) (x + 3)}{(x - 5) (x - 1)} \cdot \frac{x^{2} - 7 x + 10}{x^{2} + 2 x - 3}$

Step 4

Factor $x^{2} - 7 x + 10$ using the AC method.

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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 $10$ and whose sum is $- 7$ .

$- 5, - 2$

Step 4.2

Write the factored form using these integers.

$\frac{(x - 2) (x + 3)}{(x - 5) (x - 1)} \cdot \frac{(x - 5) (x - 2)}{x^{2} + 2 x - 3}$

Step 5

Factor $x^{2} + 2 x - 3$ 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 $- 3$ and whose sum is $2$ .

$- 1, 3$

Step 5.2

Write the factored form using these integers.

$\frac{(x - 2) (x + 3)}{(x - 5) (x - 1)} \cdot \frac{(x - 5) (x - 2)}{(x - 1) (x + 3)}$

Step 6

Cancel the common factor of $x + 3$ .

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Step 6.1

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

$\frac{(x + 3) (x - 2)}{(x - 5) (x - 1)} \cdot \frac{(x - 5) (x - 2)}{(x - 1) (x + 3)}$

Step 6.2

Factor $x + 3$ out of $(x - 1) (x + 3)$ .

$\frac{(x + 3) (x - 2)}{(x - 5) (x - 1)} \cdot \frac{(x - 5) (x - 2)}{(x + 3) (x - 1)}$

Step 6.3

Cancel the common factor.

$\frac{(x + 3) (x - 2)}{(x - 5) (x - 1)} \cdot \frac{(x - 5) (x - 2)}{(x + 3) (x - 1)}$

Step 6.4

Rewrite the expression.

$\frac{x - 2}{(x - 5) (x - 1)} \cdot \frac{(x - 5) (x - 2)}{x - 1}$

Step 7

Cancel the common factor of $x - 5$ .

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Step 7.1

Cancel the common factor.

$\frac{x - 2}{(x - 5) (x - 1)} \cdot \frac{(x - 5) (x - 2)}{x - 1}$

Step 7.2

Rewrite the expression.

$\frac{x - 2}{x - 1} \cdot \frac{x - 2}{x - 1}$

Step 8

Multiply $\frac{x - 2}{x - 1}$ by $\frac{x - 2}{x - 1}$ .

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

Step 9

Raise $x - 2$ to the power of $1$ .

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

Step 10

Raise $x - 2$ to the power of $1$ .

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

Step 11

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 12

Add $1$ and $1$ .

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

Step 13

Raise $x - 1$ to the power of $1$ .

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

Step 14

Raise $x - 1$ to the power of $1$ .

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

Step 15

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 16

Add $1$ and $1$ .

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

Step 17

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

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

Step 18

Expand $(x - 2) (x - 2)$ using the FOIL Method.

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Step 18.1

Apply the distributive property.

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

Step 18.2

Apply the distributive property.

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

Step 18.3

Apply the distributive property.

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

Step 19

Simplify and combine like terms.

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Step 19.1

Simplify each term.

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Step 19.1.1

Multiply $x$ by $x$ .

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

Step 19.1.2

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

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

Step 19.1.3

Multiply $- 2$ by $- 2$ .

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

Step 19.2

Subtract $2 x$ from $- 2 x$ .

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

Step 20

Split the fraction $\frac{x^{2} - 4 x + 4}{{(x - 1)}^{2}}$ into two fractions.

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

Step 21

Split the fraction $\frac{x^{2} - 4 x}{{(x - 1)}^{2}}$ into two fractions.

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

Step 22

Move the negative in front of the fraction.

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