Simplify 3/(y^2+y-12)-2/(y^2+6y+8)

Solution:

Step 1: Break down each term

Step 1.1: Factor $y^2+y-12$ by the AC method

Step 1.1.1: Identify two integers with a product of $-12$ and a sum of $1$. These are $-3$ and $4$.

Step 1.1.2: Express the factored form: $\frac{3}{(y-3)(y+4)}-\frac{2}{y^2+6y+8}$

Step 1.2: Factor $y^2+6y+8$ by the AC method

Step 1.2.1: Find two integers with a product of $8$ and a sum of $6$. These are $2$ and $4$.

Step 1.2.2: Write the factored form: $\frac{3}{(y-3)(y+4)}-\frac{2}{(y+2)(y+4)}$

Step 2: Adjust the first term to have a common denominator

Multiply $\frac{3}{(y-3)(y+4)}$ by $\frac{y+2}{y+2}$.

Step 3: Adjust the second term to have a common denominator

Multiply $-\frac{2}{(y+2)(y+4)}$ by $\frac{y-3}{y-3}$.

Step 4: Combine terms with a common denominator

Step 4.1: Multiply $\frac{3}{(y-3)(y+4)}$ by $\frac{y+2}{y+2}$.

Step 4.2: Multiply $\frac{2}{(y+2)(y+4)}$ by $\frac{y-3}{y-3}$.

Step 4.3: Arrange the common denominator $(y-3)(y+4)(y+2)$.

Step 5: Combine the numerators over the common denominator

$\frac{3(y+2)-2(y-3)}{(y+2)(y+4)(y-3)}$

Step 6: Simplify the numerator

Step 6.1: Distribute $3$ and $-2$ across their respective binomials.

Step 6.2: Multiply $3$ by $2$.

Step 6.3: Distribute $-2$ across the binomial $(y-3)$.

Step 6.4: Multiply $-2$ by $-3$.

Step 6.5: Combine like terms $3y-2y$.

Step 6.6: Add the constants $6+6$.

The final simplified form is $\frac{y+12}{(y+2)(y+4)(y-3)}$.

Knowledge Notes:

1. Factoring Quadratic Equations: The process of breaking down a quadratic equation into the product of two binomials. The AC method involves finding two numbers that multiply to give the product of the coefficient of $x^2$ (A) and the constant term (C), and add up to the coefficient of $x$ (B).

2. Common Denominator: When combining fractions, they must have the same denominator. To achieve this, each fraction can be multiplied by an appropriate form of 1 (a fraction where the numerator is equal to the denominator) that does not change the value of the fraction but gives it the desired common denominator.

3. Distributive Property: This property states that $a(b+c)=ab+ac$. It is used to simplify expressions by eliminating parentheses.

4. Combining Like Terms: This involves adding or subtracting terms that have the same variable raised to the same power. It simplifies expressions to a more manageable form.

5. Simplifying Expressions: The process of performing all possible simplifications, including distributing, combining like terms, and canceling factors when possible, to achieve the simplest form of an expression.