Algebra Examples

$\frac{x^{2} + 5 x + 6}{x^{2}}$

Step 1

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

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Step 1.1

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 $5$ .

$2, 3$

Step 1.2

Write the factored form using these integers.

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

Step 2

Expand $(x + 2) (x + 3)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x$ by $x$ .

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

Step 3.1.2

Move $3$ to the left of $x$ .

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

Step 3.1.3

Multiply $2$ by $3$ .

$\frac{x^{2} + 3 x + 2 x + 6}{x^{2}}$

Step 3.2

Add $3 x$ and $2 x$ .

$\frac{x^{2} + 5 x + 6}{x^{2}}$

Step 4

Split the fraction $\frac{x^{2} + 5 x + 6}{x^{2}}$ into two fractions.

$\frac{x^{2} + 5 x}{x^{2}} + \frac{6}{x^{2}}$

Step 5

Split the fraction $\frac{x^{2} + 5 x}{x^{2}}$ into two fractions.

$\frac{x^{2}}{x^{2}} + \frac{5 x}{x^{2}} + \frac{6}{x^{2}}$

Step 6

Cancel the common factor of $x^{2}$ .

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Step 6.1

Cancel the common factor.

$\frac{x^{2}}{x^{2}} + \frac{5 x}{x^{2}} + \frac{6}{x^{2}}$

Step 6.2

Rewrite the expression.

$1 + \frac{5 x}{x^{2}} + \frac{6}{x^{2}}$

Step 7

Cancel the common factor of $x$ and $x^{2}$ .

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Step 7.1

Factor $x$ out of $5 x$ .

$1 + \frac{x \cdot 5}{x^{2}} + \frac{6}{x^{2}}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

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

$1 + \frac{x \cdot 5}{x \cdot x} + \frac{6}{x^{2}}$

Step 7.2.2

Cancel the common factor.

$1 + \frac{x \cdot 5}{x \cdot x} + \frac{6}{x^{2}}$

Step 7.2.3

Rewrite the expression.

$1 + \frac{5}{x} + \frac{6}{x^{2}}$