Algebra Examples

$\frac{r + 3 s}{s + r} - \frac{3 s^{2}}{s^{2} - r^{2}} + \frac{r}{s - r}$

Step 1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = s$ and $b = r$ .

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

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{r + 3 s}{s + r}$ by $\frac{s - r}{s - r}$ .

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

Apply the distributive property.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Simplify and combine like terms.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 4.2.1.2

Multiply $r$ by $r$ by adding the exponents.

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Step 4.2.1.2.1

Move $r$ .

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

Step 4.2.1.2.2

Multiply $r$ by $r$ .

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

Step 4.2.1.3

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

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Step 4.2.1.3.1

Move $s$ .

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

Step 4.2.1.3.2

Multiply $s$ by $s$ .

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

Step 4.2.1.4

Rewrite using the commutative property of multiplication.

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

Step 4.2.1.5

Multiply $3$ by $- 1$ .

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

Step 4.2.2

Subtract $3 s r$ from $r s$ .

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Step 4.2.2.1

Move $s$ .

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

Step 4.2.2.2

Subtract $3 r s$ from $r s$ .

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

Step 4.3

Subtract $3 s^{2}$ from $3 s^{2}$ .

$\frac{- r^{2} - 2 r s + 0}{(s + r) (s - r)} + \frac{r}{s - r}$

Step 4.4

Add $- r^{2} - 2 r s$ and $0$ .

$\frac{- r^{2} - 2 r s}{(s + r) (s - r)} + \frac{r}{s - r}$

Step 4.5

Factor $r$ out of $- r^{2} - 2 r s$ .

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Step 4.5.1

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

$\frac{r (- r) - 2 r s}{(s + r) (s - r)} + \frac{r}{s - r}$

Step 4.5.2

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

$\frac{r (- r) + r (- 2 s)}{(s + r) (s - r)} + \frac{r}{s - r}$

Step 4.5.3

Factor $r$ out of $r (- r) + r (- 2 s)$ .

$\frac{r (- r - 2 s)}{(s + r) (s - r)} + \frac{r}{s - r}$

Step 5

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

$\frac{r (- r - 2 s)}{(s + r) (s - r)} + \frac{r}{s - r} \cdot \frac{s + r}{s + r}$

Step 6

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

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Step 6.1

Multiply $\frac{r}{s - r}$ by $\frac{s + r}{s + r}$ .

$\frac{r (- r - 2 s)}{(s + r) (s - r)} + \frac{r (s + r)}{(s - r) (s + r)}$

Step 6.2

Reorder the factors of $(s - r) (s + r)$ .

$\frac{r (- r - 2 s)}{(s + r) (s - r)} + \frac{r (s + r)}{(s + r) (s - r)}$

Step 7

Combine the numerators over the common denominator.

$\frac{r (- r - 2 s) + r (s + r)}{(s + r) (s - r)}$

Step 8

Simplify the numerator.

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Step 8.1

Factor $r$ out of $r (- r - 2 s) + r (s + r)$ .

$\frac{r (- r - 2 s + s + r)}{(s + r) (s - r)}$

Step 8.2

Add $- r$ and $r$ .

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

Step 8.3

Add $- 2 s + s$ and $0$ .

$\frac{r (- 2 s + s)}{(s + r) (s - r)}$

Step 8.4

Add $- 2 s$ and $s$ .

$\frac{r \cdot - 1 s}{(s + r) (s - r)}$

Step 8.5

Factor out negative.

$\frac{- r s}{(s + r) (s - r)}$

Step 9

Move the negative in front of the fraction.

$- \frac{r s}{(s + r) (s - r)}$