Simplify 3/(y^2+y-12)-2/(y^2+6y+8)
The given question requires the simplification of a mathematical expression involving the subtraction of two rational expressions (fractions), each with a quadratic polynomial in the denominator. The first rational expression has 3 as its numerator and a quadratic trinomial y^2 + y - 12 as its denominator. The second rational expression has 2 as its numerator and another quadratic trinomial y^2 + 6y + 8 as its denominator. The task is to perform the simplification by finding a common denominator for both fractions and then combining them appropriately to result in a simplified form. Simplification may include factoring the quadratic expressions and canceling out common factors, if any. The final expression should be a single rational expression with a simplified numerator and denominator.
$\frac{3}{y^{2} + y - 12} - \frac{2}{y^{2} + 6 y + 8}$
Answer
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:
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).
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.
Distributive Property: This property states that $a(b + c) = ab + ac$. It is used to simplify expressions by eliminating parentheses.
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.
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.
