Algebra Examples

$y - 1 - \frac{5}{y + 3}$

Step 1

To write $y$ as a fraction with a common denominator, multiply by $\frac{y + 3}{y + 3}$ .

$\frac{y (y + 3)}{y + 3} - \frac{5}{y + 3} - 1$

Step 2

Combine the numerators over the common denominator.

$\frac{y (y + 3) - 5}{y + 3} - 1$

Step 3

Simplify the numerator.

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Step 3.1

Apply the distributive property.

$\frac{y \cdot y + y \cdot 3 - 5}{y + 3} - 1$

Step 3.2

Multiply $y$ by $y$ .

$\frac{y^{2} + y \cdot 3 - 5}{y + 3} - 1$

Step 3.3

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

$\frac{y^{2} + 3 y - 5}{y + 3} - 1$

Step 4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{y + 3}{y + 3}$ .

$\frac{y^{2} + 3 y - 5}{y + 3} - 1 \cdot \frac{y + 3}{y + 3}$

Step 5

Combine $- 1$ and $\frac{y + 3}{y + 3}$ .

$\frac{y^{2} + 3 y - 5}{y + 3} + \frac{- (y + 3)}{y + 3}$

Step 6

Combine the numerators over the common denominator.

$\frac{y^{2} + 3 y - 5 - (y + 3)}{y + 3}$

Step 7

Simplify the numerator.

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Step 7.1

Apply the distributive property.

$\frac{y^{2} + 3 y - 5 - y - 1 \cdot 3}{y + 3}$

Step 7.2

Multiply $- 1$ by $3$ .

$\frac{y^{2} + 3 y - 5 - y - 3}{y + 3}$

Step 7.3

Subtract $y$ from $3 y$ .

$\frac{y^{2} + 2 y - 5 - 3}{y + 3}$

Step 7.4

Subtract $3$ from $- 5$ .

$\frac{y^{2} + 2 y - 8}{y + 3}$

Step 7.5

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

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Step 7.5.1

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 $- 8$ and whose sum is $2$ .

$- 2, 4$

Step 7.5.2

Write the factored form using these integers.

$\frac{(y - 2) (y + 4)}{y + 3}$