Algebra Examples

$\frac{x + 2}{\frac{x^{2} + 2 x - 3}{\frac{x + 2}{x^{2} - x}}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Factor $x$ out of $- x$ .

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

Step 2.3

Factor $x$ out of $x \cdot x + x \cdot - 1$ .

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

Step 3

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

$- 1, 3$

Step 3.2

Write the factored form using these integers.

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

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

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Step 5.1

Multiply $\frac{x + 2}{x (x - 1)}$ by $\frac{1}{(x - 1) (x + 3)}$ .

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

Add $1$ and $1$ .

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

Step 6

Multiply $x + 2$ by $\frac{x + 2}{x {(x - 1)}^{2} (x + 3)}$ .

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

Step 7

Simplify the numerator.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

Add $1$ and $1$ .

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