Algebra Examples

$\frac{s}{3 s + 2} + \frac{s + 3}{2 s - 4} = \frac{- 2 s}{3 s^{2} - 4 s - 4}$

Step 1

Subtract $\frac{- 2 s}{3 s^{2} - 4 s - 4}$ from both sides of the equation.

$\frac{s}{3 s + 2} + \frac{s + 3}{2 s - 4} - \frac{- 2 s}{3 s^{2} - 4 s - 4} = 0$

Step 2

Simplify $\frac{s}{3 s + 2} + \frac{s + 3}{2 s - 4} - \frac{- 2 s}{3 s^{2} - 4 s - 4}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Factor $2$ out of $2 s - 4$ .

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Step 2.1.1.1

Factor $2$ out of $2 s$ .

$\frac{s}{3 s + 2} + \frac{s + 3}{2 (s) - 4} - \frac{- 2 s}{3 s^{2} - 4 s - 4} = 0$

Step 2.1.1.2

Factor $2$ out of $- 4$ .

$\frac{s}{3 s + 2} + \frac{s + 3}{2 s + 2 \cdot - 2} - \frac{- 2 s}{3 s^{2} - 4 s - 4} = 0$

Step 2.1.1.3

Factor $2$ out of $2 s + 2 \cdot - 2$ .

$\frac{s}{3 s + 2} + \frac{s + 3}{2 (s - 2)} - \frac{- 2 s}{3 s^{2} - 4 s - 4} = 0$

Step 2.1.2

Factor by grouping.

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Step 2.1.2.1

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 $a \cdot c = 3 \cdot - 4 = - 12$ and whose sum is $b = - 4$ .

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Step 2.1.2.1.1

Factor $- 4$ out of $- 4 s$ .

$\frac{s}{3 s + 2} + \frac{s + 3}{2 (s - 2)} - \frac{- 2 s}{3 s^{2} - 4 (s) - 4} = 0$

Step 2.1.2.1.2

Rewrite $- 4$ as $2$ plus $- 6$

$\frac{s}{3 s + 2} + \frac{s + 3}{2 (s - 2)} - \frac{- 2 s}{3 s^{2} + (2 - 6) s - 4} = 0$

Step 2.1.2.1.3

Apply the distributive property.

$\frac{s}{3 s + 2} + \frac{s + 3}{2 (s - 2)} - \frac{- 2 s}{3 s^{2} + 2 s - 6 s - 4} = 0$

Step 2.1.2.2

Factor out the greatest common factor from each group.

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Step 2.1.2.2.1

Group the first two terms and the last two terms.

$\frac{s}{3 s + 2} + \frac{s + 3}{2 (s - 2)} - \frac{- 2 s}{(3 s^{2} + 2 s) - 6 s - 4} = 0$

Step 2.1.2.2.2

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

$\frac{s}{3 s + 2} + \frac{s + 3}{2 (s - 2)} - \frac{- 2 s}{s (3 s + 2) - 2 (3 s + 2)} = 0$

Step 2.1.2.3

Factor the polynomial by factoring out the greatest common factor, $3 s + 2$ .

$\frac{s}{3 s + 2} + \frac{s + 3}{2 (s - 2)} - \frac{- 2 s}{(3 s + 2) (s - 2)} = 0$

Step 2.1.3

Move the negative in front of the fraction.

$\frac{s}{3 s + 2} + \frac{s + 3}{2 (s - 2)} - - \frac{2 s}{(3 s + 2) (s - 2)} = 0$

Step 2.1.4

Multiply $- - \frac{2 s}{(3 s + 2) (s - 2)}$ .

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Step 2.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{s}{3 s + 2} + \frac{s + 3}{2 (s - 2)} + 1 \frac{2 s}{(3 s + 2) (s - 2)} = 0$

Step 2.1.4.2

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

$\frac{s}{3 s + 2} + \frac{s + 3}{2 (s - 2)} + \frac{2 s}{(3 s + 2) (s - 2)} = 0$

Step 2.2

To write $\frac{s}{3 s + 2}$ as a fraction with a common denominator, multiply by $\frac{2 (s - 2)}{2 (s - 2)}$ .

$\frac{s}{3 s + 2} \cdot \frac{2 (s - 2)}{2 (s - 2)} + \frac{s + 3}{2 (s - 2)} + \frac{2 s}{(3 s + 2) (s - 2)} = 0$

Step 2.3

To write $\frac{s + 3}{2 (s - 2)}$ as a fraction with a common denominator, multiply by $\frac{3 s + 2}{3 s + 2}$ .

$\frac{s}{3 s + 2} \cdot \frac{2 (s - 2)}{2 (s - 2)} + \frac{s + 3}{2 (s - 2)} \cdot \frac{3 s + 2}{3 s + 2} + \frac{2 s}{(3 s + 2) (s - 2)} = 0$

Step 2.4

Write each expression with a common denominator of $(3 s + 2) \cdot 2 (s - 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

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

$\frac{s (2 (s - 2))}{(3 s + 2) (2 (s - 2))} + \frac{s + 3}{2 (s - 2)} \cdot \frac{3 s + 2}{3 s + 2} + \frac{2 s}{(3 s + 2) (s - 2)} = 0$

Step 2.4.2

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

$\frac{s (2 (s - 2))}{(3 s + 2) (2 (s - 2))} + \frac{(s + 3) (3 s + 2)}{2 (s - 2) (3 s + 2)} + \frac{2 s}{(3 s + 2) (s - 2)} = 0$

Step 2.4.3

Reorder the factors of $(3 s + 2) (2 (s - 2))$ .

$\frac{s (2 (s - 2))}{2 (3 s + 2) (s - 2)} + \frac{(s + 3) (3 s + 2)}{2 (s - 2) (3 s + 2)} + \frac{2 s}{(3 s + 2) (s - 2)} = 0$

Step 2.4.4

Reorder the factors of $2 (s - 2) (3 s + 2)$ .

$\frac{s (2 (s - 2))}{2 (3 s + 2) (s - 2)} + \frac{(s + 3) (3 s + 2)}{2 (3 s + 2) (s - 2)} + \frac{2 s}{(3 s + 2) (s - 2)} = 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{s (2 (s - 2)) + (s + 3) (3 s + 2)}{2 (3 s + 2) (s - 2)} + \frac{2 s}{(3 s + 2) (s - 2)} = 0$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Rewrite using the commutative property of multiplication.

$\frac{2 s (s - 2) + (s + 3) (3 s + 2)}{2 (3 s + 2) (s - 2)} + \frac{2 s}{(3 s + 2) (s - 2)} = 0$

Step 2.6.2

Apply the distributive property.

$\frac{2 s \cdot s + 2 s \cdot - 2 + (s + 3) (3 s + 2)}{2 (3 s + 2) (s - 2)} + \frac{2 s}{(3 s + 2) (s - 2)} = 0$

Step 2.6.3

Multiply $s$ by $s$ by adding the exponents.

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Step 2.6.3.1

Move $s$ .

$\frac{2 (s \cdot s) + 2 s \cdot - 2 + (s + 3) (3 s + 2)}{2 (3 s + 2) (s - 2)} + \frac{2 s}{(3 s + 2) (s - 2)} = 0$

Step 2.6.3.2

Multiply $s$ by $s$ .

$\frac{2 s^{2} + 2 s \cdot - 2 + (s + 3) (3 s + 2)}{2 (3 s + 2) (s - 2)} + \frac{2 s}{(3 s + 2) (s - 2)} = 0$

Step 2.6.4

Multiply $- 2$ by $2$ .

$\frac{2 s^{2} - 4 s + (s + 3) (3 s + 2)}{2 (3 s + 2) (s - 2)} + \frac{2 s}{(3 s + 2) (s - 2)} = 0$

Step 2.6.5

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

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Step 2.6.5.1

Apply the distributive property.

$\frac{2 s^{2} - 4 s + s (3 s + 2) + 3 (3 s + 2)}{2 (3 s + 2) (s - 2)} + \frac{2 s}{(3 s + 2) (s - 2)} = 0$

Step 2.6.5.2

Apply the distributive property.

$\frac{2 s^{2} - 4 s + s (3 s) + s \cdot 2 + 3 (3 s + 2)}{2 (3 s + 2) (s - 2)} + \frac{2 s}{(3 s + 2) (s - 2)} = 0$

Step 2.6.5.3

Apply the distributive property.

$\frac{2 s^{2} - 4 s + s (3 s) + s \cdot 2 + 3 (3 s) + 3 \cdot 2}{2 (3 s + 2) (s - 2)} + \frac{2 s}{(3 s + 2) (s - 2)} = 0$

Step 2.6.6

Simplify and combine like terms.

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Step 2.6.6.1

Simplify each term.

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Step 2.6.6.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 s^{2} - 4 s + 3 s \cdot s + s \cdot 2 + 3 (3 s) + 3 \cdot 2}{2 (3 s + 2) (s - 2)} + \frac{2 s}{(3 s + 2) (s - 2)} = 0$

Step 2.6.6.1.2

Multiply $s$ by $s$ by adding the exponents.

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Step 2.6.6.1.2.1

Move $s$ .

$\frac{2 s^{2} - 4 s + 3 (s \cdot s) + s \cdot 2 + 3 (3 s) + 3 \cdot 2}{2 (3 s + 2) (s - 2)} + \frac{2 s}{(3 s + 2) (s - 2)} = 0$

Step 2.6.6.1.2.2

Multiply $s$ by $s$ .

$\frac{2 s^{2} - 4 s + 3 s^{2} + s \cdot 2 + 3 (3 s) + 3 \cdot 2}{2 (3 s + 2) (s - 2)} + \frac{2 s}{(3 s + 2) (s - 2)} = 0$

Step 2.6.6.1.3

Move $2$ to the left of $s$ .

$\frac{2 s^{2} - 4 s + 3 s^{2} + 2 \cdot s + 3 (3 s) + 3 \cdot 2}{2 (3 s + 2) (s - 2)} + \frac{2 s}{(3 s + 2) (s - 2)} = 0$

Step 2.6.6.1.4

Multiply $3$ by $3$ .

$\frac{2 s^{2} - 4 s + 3 s^{2} + 2 s + 9 s + 3 \cdot 2}{2 (3 s + 2) (s - 2)} + \frac{2 s}{(3 s + 2) (s - 2)} = 0$

Step 2.6.6.1.5

Multiply $3$ by $2$ .

$\frac{2 s^{2} - 4 s + 3 s^{2} + 2 s + 9 s + 6}{2 (3 s + 2) (s - 2)} + \frac{2 s}{(3 s + 2) (s - 2)} = 0$

Step 2.6.6.2

Add $2 s$ and $9 s$ .

$\frac{2 s^{2} - 4 s + 3 s^{2} + 11 s + 6}{2 (3 s + 2) (s - 2)} + \frac{2 s}{(3 s + 2) (s - 2)} = 0$

Step 2.6.7

Add $2 s^{2}$ and $3 s^{2}$ .

$\frac{5 s^{2} - 4 s + 11 s + 6}{2 (3 s + 2) (s - 2)} + \frac{2 s}{(3 s + 2) (s - 2)} = 0$

Step 2.6.8

Add $- 4 s$ and $11 s$ .

$\frac{5 s^{2} + 7 s + 6}{2 (3 s + 2) (s - 2)} + \frac{2 s}{(3 s + 2) (s - 2)} = 0$

Step 2.7

To write $\frac{2 s}{(3 s + 2) (s - 2)}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{5 s^{2} + 7 s + 6}{2 (3 s + 2) (s - 2)} + \frac{2 s}{(3 s + 2) (s - 2)} \cdot \frac{2}{2} = 0$

Step 2.8

Write each expression with a common denominator of $2 (3 s + 2) (s - 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.8.1

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

$\frac{5 s^{2} + 7 s + 6}{2 (3 s + 2) (s - 2)} + \frac{2 s \cdot 2}{(3 s + 2) (s - 2) \cdot 2} = 0$

Step 2.8.2

Reorder the factors of $(3 s + 2) (s - 2) \cdot 2$ .

$\frac{5 s^{2} + 7 s + 6}{2 (3 s + 2) (s - 2)} + \frac{2 s \cdot 2}{2 (3 s + 2) (s - 2)} = 0$

Step 2.9

Combine the numerators over the common denominator.

$\frac{5 s^{2} + 7 s + 6 + 2 s \cdot 2}{2 (3 s + 2) (s - 2)} = 0$

Step 2.10

Simplify the numerator.

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Step 2.10.1

Multiply $2$ by $2$ .

$\frac{5 s^{2} + 7 s + 6 + 4 s}{2 (3 s + 2) (s - 2)} = 0$

Step 2.10.2

Add $7 s$ and $4 s$ .

$\frac{5 s^{2} + 11 s + 6}{2 (3 s + 2) (s - 2)} = 0$

Step 2.10.3

Factor by grouping.

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Step 2.10.3.1

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 $a \cdot c = 5 \cdot 6 = 30$ and whose sum is $b = 11$ .

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Step 2.10.3.1.1

Factor $11$ out of $11 s$ .

$\frac{5 s^{2} + 11 (s) + 6}{2 (3 s + 2) (s - 2)} = 0$

Step 2.10.3.1.2

Rewrite $11$ as $5$ plus $6$

$\frac{5 s^{2} + (5 + 6) s + 6}{2 (3 s + 2) (s - 2)} = 0$

Step 2.10.3.1.3

Apply the distributive property.

$\frac{5 s^{2} + 5 s + 6 s + 6}{2 (3 s + 2) (s - 2)} = 0$

Step 2.10.3.2

Factor out the greatest common factor from each group.

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Step 2.10.3.2.1

Group the first two terms and the last two terms.

$\frac{(5 s^{2} + 5 s) + 6 s + 6}{2 (3 s + 2) (s - 2)} = 0$

Step 2.10.3.2.2

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

$\frac{5 s (s + 1) + 6 (s + 1)}{2 (3 s + 2) (s - 2)} = 0$

Step 2.10.3.3

Factor the polynomial by factoring out the greatest common factor, $s + 1$ .

$\frac{(s + 1) (5 s + 6)}{2 (3 s + 2) (s - 2)} = 0$

Step 3

Set the numerator equal to zero.

$(s + 1) (5 s + 6) = 0$

Step 4

Solve the equation for $s$ .

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Step 4.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$s + 1 = 0$

$5 s + 6 = 0$

Step 4.2

Set $s + 1$ equal to $0$ and solve for $s$ .

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Step 4.2.1

Set $s + 1$ equal to $0$ .

$s + 1 = 0$

Step 4.2.2

Subtract $1$ from both sides of the equation.

$s = - 1$

Step 4.3

Set $5 s + 6$ equal to $0$ and solve for $s$ .

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Step 4.3.1

Set $5 s + 6$ equal to $0$ .

$5 s + 6 = 0$

Step 4.3.2

Solve $5 s + 6 = 0$ for $s$ .

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Step 4.3.2.1

Subtract $6$ from both sides of the equation.

$5 s = - 6$

Step 4.3.2.2

Divide each term in $5 s = - 6$ by $5$ and simplify.

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Step 4.3.2.2.1

Divide each term in $5 s = - 6$ by $5$ .

$\frac{5 s}{5} = \frac{- 6}{5}$

Step 4.3.2.2.2

Simplify the left side.

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Step 4.3.2.2.2.1

Cancel the common factor of $5$ .

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Step 4.3.2.2.2.1.1

Cancel the common factor.

$\frac{5 s}{5} = \frac{- 6}{5}$

Step 4.3.2.2.2.1.2

Divide $s$ by $1$ .

$s = \frac{- 6}{5}$

Step 4.3.2.2.3

Simplify the right side.

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Step 4.3.2.2.3.1

Move the negative in front of the fraction.

$s = - \frac{6}{5}$

Step 4.4

The final solution is all the values that make $(s + 1) (5 s + 6) = 0$ true.

$s = - 1, - \frac{6}{5}$