Finite Math Examples

$s = \frac{a - a r^{2}}{1 - r}$

Step 1

Rewrite the equation as $\frac{a - a r^{2}}{1 - r} = s$ .

$\frac{a - a r^{2}}{1 - r} = s$

Step 2

Multiply both sides by $1 - r$ .

$\frac{a - a r^{2}}{1 - r} (1 - r) = s (1 - r)$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Simplify the left side.

Tap for more steps...

Step 3.1.1

Simplify $\frac{a - a r^{2}}{1 - r} (1 - r)$ .

Tap for more steps...

Step 3.1.1.1

Cancel the common factor of $1 - r$ .

Tap for more steps...

Step 3.1.1.1.1

Cancel the common factor.

$\frac{a - a r^{2}}{1 - r} (1 - r) = s (1 - r)$

Step 3.1.1.1.2

Rewrite the expression.

$a - a r^{2} = s (1 - r)$

Step 3.1.1.2

Reorder $a$ and $- a r^{2}$ .

$- a r^{2} + a = s (1 - r)$

Step 3.2

Simplify the right side.

Tap for more steps...

Step 3.2.1

Simplify $s (1 - r)$ .

Tap for more steps...

Step 3.2.1.1

Apply the distributive property.

$- a r^{2} + a = s \cdot 1 + s (- r)$

Step 3.2.1.2

Simplify the expression.

Tap for more steps...

Step 3.2.1.2.1

Multiply $s$ by $1$ .

$- a r^{2} + a = s + s (- r)$

Step 3.2.1.2.2

Rewrite using the commutative property of multiplication.

$- a r^{2} + a = s - s r$

Step 3.2.1.2.3

Move $s$ .

$- a r^{2} + a = s - r s$

Step 3.2.1.2.4

Reorder $s$ and $- r s$ .

$- a r^{2} + a = - r s + s$

Step 4

Solve for $a$ .

Tap for more steps...

Step 4.1

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

Tap for more steps...

Step 4.1.1

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

$a (- 1 r^{2}) + a = - r s + s$

Step 4.1.2

Raise $a$ to the power of $1$ .

$a (- 1 r^{2}) + a = - r s + s$

Step 4.1.3

Factor $a$ out of $a^{1}$ .

$a (- 1 r^{2}) + a \cdot 1 = - r s + s$

Step 4.1.4

Factor $a$ out of $a (- 1 r^{2}) + a \cdot 1$ .

$a (- 1 r^{2} + 1) = - r s + s$

Step 4.2

Rewrite $1$ as $1^{2}$ .

$a (- 1 r^{2} + 1^{2}) = - r s + s$

Step 4.3

Reorder $- 1 r^{2}$ and $1^{2}$ .

$a (1^{2} - 1 r^{2}) = - r s + s$

Step 4.4

Factor.

Tap for more steps...

Step 4.4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = r$ .

$a ((1 + r) (1 - r)) = - r s + s$

Step 4.4.2

Remove unnecessary parentheses.

$a (1 + r) (1 - r) = - r s + s$

Step 4.5

Divide each term in $a (1 + r) (1 - r) = - r s + s$ by $(1 + r) (1 - r)$ and simplify.

Tap for more steps...

Step 4.5.1

Divide each term in $a (1 + r) (1 - r) = - r s + s$ by $(1 + r) (1 - r)$ .

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

Step 4.5.2

Simplify the left side.

Tap for more steps...

Step 4.5.2.1

Cancel the common factor of $1 + r$ .

Tap for more steps...

Step 4.5.2.1.1

Cancel the common factor.

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

Step 4.5.2.1.2

Rewrite the expression.

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

Step 4.5.2.2

Cancel the common factor of $1 - r$ .

Tap for more steps...

Step 4.5.2.2.1

Cancel the common factor.

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

Step 4.5.2.2.2

Divide $a$ by $1$ .

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

Step 4.5.3

Simplify the right side.

Tap for more steps...

Step 4.5.3.1

Move the negative in front of the fraction.

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