Solve for n (n^2-n-6)/(n^2)-(2n+12)/n=(n-6)/(2n)

Solution:

Step:1

Break down each term into its factors.

Step:1.1

Employ the AC method to factor $n^{2} - n - 6$.

Step:1.1.1

Identify two integers whose product equals $c$ and sum equals $b$ for the quadratic form $ax^{2} + bx + c$. Here, find integers for product $-6$ and sum $-1$. They are $-3$ and $2$.

Step:1.1.2

Express the factored form using these integers: $\frac{(n - 3)(n + 2)}{n^{2}} - \frac{2n + 12}{n} = \frac{n - 6}{2n}$.

Step:1.2

Extract the common factor of $2$ from $2n + 12$.

Step:1.2.1

Factor out $2$ from $2n$: $\frac{(n - 3)(n + 2)}{n^{2}} - \frac{2(n + 6)}{n} = \frac{n - 6}{2n}$.

Step:1.2.2

Factor out $2$ from $12$: $\frac{(n - 3)(n + 2)}{n^{2}} - \frac{2(n + 6)}{n} = \frac{n - 6}{2n}$.

Step:1.2.3

The factored form is now: $\frac{(n - 3)(n + 2)}{n^{2}} - \frac{2(n + 6)}{n} = \frac{n - 6}{2n}$.

Step:2

Identify the least common denominator (LCD) of all terms.

Step:2.1

The LCD is equivalent to the least common multiple (LCM) of the denominators: $n^{2}, n, 2n$.

Step:2.2

To find the LCM of $n^{2}, n, 2n$, consider both the numerical and variable parts. For numbers, find the LCM of $1, 1, 2$. For variables, consider $n^{2}, n, n$.

Step:2.3

The LCM is the smallest number divisible by all. List prime factors and multiply each factor by the highest power it occurs.

Step:2.4

The number $1$ is not prime.

Step:2.5

The number $2$ is prime.

Step:2.6

The LCM of $1, 1, 2$ is $2$.

Step:2.7

For $n^{2}$, the factors are $n \cdot n$.

Step:2.8

For $n$, the factor is $n$ itself.

Step:2.9

The LCM of $n^{2}, n, n$ is $n \cdot n$.

Step:2.10

Multiply $n$ by $n$: $n^{2}$.

Step:2.11

The LCM for $n^{2}, n, 2n$ is $2n^{2}$.

Step:3

Clear the fractions by multiplying each term by $2n^{2}$.

Step:3.1

Multiply the entire equation by $2n^{2}$.

Step:3.2

Simplify the left side.

Step:3.2.1

Simplify each term.

Step:3.2.1.1

Use the commutative property: $2(n - 3)(n + 2) - 2(n + 6)(2n) = (n - 6)(2n^{2})$.

Step:3.2.1.2

Combine the terms: $2(n - 3)(n + 2) - 2(n + 6)(2n) = (n - 6)(2n^{2})$.

Step:3.2.1.3

Cancel $n^{2}$: $2(n - 3)(n + 2) - 2(n + 6)(2n) = (n - 6)(2n^{2})$.

Step:3.2.1.4

Expand using FOIL: $2(n \cdot n + n \cdot 2 - 3n - 6) - 2(n + 6)(2n) = (n - 6)(2n^{2})$.

Step:3.2.1.5

Combine like terms: $2n^{2} - 2n - 12 - 2(n + 6)(2n) = (n - 6)(2n^{2})$.

Step:3.3

Simplify the right side.

Step:3.3.1

Use commutative property: $-2n^{2} - 26n - 12 = (n - 6)n^{2}$.

Step:3.3.2

Cancel the common factor of $2$: $-2n^{2} - 26n - 12 = (n - 6)n$.

Step:3.3.3

Cancel the common factor of $n$: $-2n^{2} - 26n - 12 = (n - 6)n$.

Step:3.3.4

Apply distributive property: $-2n^{2} - 26n - 12 = n^{2} - 6n$.

Step:4

Resolve the equation.

Step:4.1

Move all $n$ terms to one side: $-3n^{2} - 20n - 12 = 0$.

Step:4.2

Factor the quadratic equation.

Step:4.2.1

Factor out $-1$: $-(3n^{2} + 20n + 12) = 0$.

Step:4.2.2

Factor by grouping: $-(3n + 2)(n + 6) = 0$.

Step:4.3

Set each factor equal to zero: $3n + 2 = 0$ and $n + 6 = 0$.

Step:4.4

Solve for $n$: $n = -\frac{2}{3}$ and $n = -6$.

Step:5

Express the solution in various forms.

Exact Form: $n = -\frac{2}{3}, -6$ Decimal Form: $n = -0.666\ldots, -6$

Knowledge Notes:

1. Factoring Quadratics: The process of breaking down a quadratic expression into a product of two binomials.

2. AC Method: A technique used to factor quadratics where the product of the leading coefficient and the constant term (AC) is considered to find a pair of numbers that sum up to the middle coefficient (B).

3. Least Common Denominator (LCD): The smallest common multiple of the denominators of a set of fractions.

4. Least Common Multiple (LCM): The smallest number that is a multiple of two or more numbers.

5. Distributive Property: A property that allows one to multiply a sum by multiplying each addend separately and then sum the products.

6. FOIL Method: A technique for multiplying two binomials which stands for First, Outer, Inner, Last, referring to the terms being multiplied.

7. Factoring by Grouping: A method of factoring that involves grouping terms with common factors and factoring them separately.

8. Solving Quadratic Equations: The process of finding the values of the variable that satisfy the equation. This can be done by factoring, completing the square, or using the quadratic formula.