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 $a{x}^2+bx+c$. 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: $n=-0.666\dots ,-6$

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.