Finite Math Examples

$a (n) = \frac{n + 1}{n + 3}$

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, n + 3$

Step 1.2

Remove parentheses.

$1, n + 3$

Step 1.3

The LCM of one and any expression is the expression.

$n + 3$

Step 2

Multiply each term in $a n = \frac{n + 1}{n + 3}$ by $n + 3$ to eliminate the fractions.

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Step 2.1

Multiply each term in $a n = \frac{n + 1}{n + 3}$ by $n + 3$ .

$a n (n + 3) = \frac{n + 1}{n + 3} (n + 3)$

Step 2.2

Simplify the left side.

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Step 2.2.1

Apply the distributive property.

$a n \cdot n + a n \cdot 3 = \frac{n + 1}{n + 3} (n + 3)$

Step 2.2.2

Multiply $n$ by $n$ by adding the exponents.

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Step 2.2.2.1

Move $n$ .

$a (n \cdot n) + a n \cdot 3 = \frac{n + 1}{n + 3} (n + 3)$

Step 2.2.2.2

Multiply $n$ by $n$ .

$a n^{2} + a n \cdot 3 = \frac{n + 1}{n + 3} (n + 3)$

Step 2.2.3

Move $3$ to the left of $a n$ .

$a n^{2} + 3 \cdot (a n) = \frac{n + 1}{n + 3} (n + 3)$

Step 2.2.4

Remove parentheses.

$a n^{2} + 3 a n = \frac{n + 1}{n + 3} (n + 3)$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $n + 3$ .

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Step 2.3.1.1

Cancel the common factor.

$a n^{2} + 3 a n = \frac{n + 1}{n + 3} (n + 3)$

Step 2.3.1.2

Rewrite the expression.

$a n^{2} + 3 a n = n + 1$

Step 3

Solve the equation.

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Step 3.1

Subtract $n$ from both sides of the equation.

$a n^{2} + 3 a n - n = 1$

Step 3.2

Subtract $1$ from both sides of the equation.

$a n^{2} + 3 a n - n - 1 = 0$

Step 3.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.4

Substitute the values $a = a$ , $b = 3 a - 1$ , and $c = - 1$ into the quadratic formula and solve for $n$ .

$\frac{- (3 a - 1) \pm \sqrt{{(3 a - 1)}^{2} - 4 \cdot (a \cdot - 1)}}{2 a}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Apply the distributive property.

$n = \frac{- (3 a) + 1 \pm \sqrt{{(3 a - 1)}^{2} - 4 \cdot a \cdot - 1}}{2 a}$

Step 3.5.2

Multiply $3$ by $- 1$ .

$n = \frac{- 3 a + 1 \pm \sqrt{{(3 a - 1)}^{2} - 4 \cdot a \cdot - 1}}{2 a}$

Step 3.5.3

Multiply $- 1$ by $- 1$ .

$n = \frac{- 3 a + 1 \pm \sqrt{{(3 a - 1)}^{2} - 4 \cdot a \cdot - 1}}{2 a}$

Step 3.5.4

Rewrite ${(3 a - 1)}^{2}$ as $(3 a - 1) (3 a - 1)$ .

$n = \frac{- 3 a + 1 \pm \sqrt{(3 a - 1) (3 a - 1) - 4 \cdot a \cdot - 1}}{2 a}$

Step 3.5.5

Expand $(3 a - 1) (3 a - 1)$ using the FOIL Method.

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Step 3.5.5.1

Apply the distributive property.

$n = \frac{- 3 a + 1 \pm \sqrt{3 a (3 a - 1) - 1 (3 a - 1) - 4 \cdot a \cdot - 1}}{2 a}$

Step 3.5.5.2

Apply the distributive property.

$n = \frac{- 3 a + 1 \pm \sqrt{3 a (3 a) + 3 a \cdot - 1 - 1 (3 a - 1) - 4 \cdot a \cdot - 1}}{2 a}$

Step 3.5.5.3

Apply the distributive property.

$n = \frac{- 3 a + 1 \pm \sqrt{3 a (3 a) + 3 a \cdot - 1 - 1 (3 a) - 1 \cdot - 1 - 4 \cdot a \cdot - 1}}{2 a}$

Step 3.5.6

Simplify and combine like terms.

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Step 3.5.6.1

Simplify each term.

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Step 3.5.6.1.1

Rewrite using the commutative property of multiplication.

$n = \frac{- 3 a + 1 \pm \sqrt{3 \cdot (3 a \cdot a) + 3 a \cdot - 1 - 1 (3 a) - 1 \cdot - 1 - 4 \cdot a \cdot - 1}}{2 a}$

Step 3.5.6.1.2

Multiply $a$ by $a$ by adding the exponents.

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Step 3.5.6.1.2.1

Move $a$ .

$n = \frac{- 3 a + 1 \pm \sqrt{3 \cdot (3 (a \cdot a)) + 3 a \cdot - 1 - 1 (3 a) - 1 \cdot - 1 - 4 \cdot a \cdot - 1}}{2 a}$

Step 3.5.6.1.2.2

Multiply $a$ by $a$ .

$n = \frac{- 3 a + 1 \pm \sqrt{3 \cdot (3 a^{2}) + 3 a \cdot - 1 - 1 (3 a) - 1 \cdot - 1 - 4 \cdot a \cdot - 1}}{2 a}$

Step 3.5.6.1.3

Multiply $3$ by $3$ .

$n = \frac{- 3 a + 1 \pm \sqrt{9 a^{2} + 3 a \cdot - 1 - 1 (3 a) - 1 \cdot - 1 - 4 \cdot a \cdot - 1}}{2 a}$

Step 3.5.6.1.4

Multiply $- 1$ by $3$ .

$n = \frac{- 3 a + 1 \pm \sqrt{9 a^{2} - 3 a - 1 (3 a) - 1 \cdot - 1 - 4 \cdot a \cdot - 1}}{2 a}$

Step 3.5.6.1.5

Multiply $3$ by $- 1$ .

$n = \frac{- 3 a + 1 \pm \sqrt{9 a^{2} - 3 a - 3 a - 1 \cdot - 1 - 4 \cdot a \cdot - 1}}{2 a}$

Step 3.5.6.1.6

Multiply $- 1$ by $- 1$ .

$n = \frac{- 3 a + 1 \pm \sqrt{9 a^{2} - 3 a - 3 a + 1 - 4 \cdot a \cdot - 1}}{2 a}$

Step 3.5.6.2

Subtract $3 a$ from $- 3 a$ .

$n = \frac{- 3 a + 1 \pm \sqrt{9 a^{2} - 6 a + 1 - 4 \cdot a \cdot - 1}}{2 a}$

Step 3.5.7

Multiply $- 1$ by $- 4$ .

$n = \frac{- 3 a + 1 \pm \sqrt{9 a^{2} - 6 a + 1 + 4 a}}{2 a}$

Step 3.5.8

Add $- 6 a$ and $4 a$ .

$n = \frac{- 3 a + 1 \pm \sqrt{9 a^{2} - 2 a + 1}}{2 a}$

Step 3.6

The final answer is the combination of both solutions.

$n = - \frac{3 a - 1 - \sqrt{9 a^{2} - 2 a + 1}}{2 a}$

$n = - \frac{3 a - 1 + \sqrt{9 a^{2} - 2 a + 1}}{2 a}$

$n = - \frac{3 a - 1 - \sqrt{9 a^{2} - 2 a + 1}}{2 a}$

$n = - \frac{3 a - 1 + \sqrt{9 a^{2} - 2 a + 1}}{2 a}$