Math Problem Statement
Solution
To solve the given problem of adding the rational expressions with different quadratic denominators, let's go through the steps carefully.
Problem
Add the following rational expressions:
Step 1: Factor the Denominators
We start by factoring each quadratic expression in the denominator if possible.
-
For :
- We need two numbers that multiply to and add up to .
- The factors and satisfy these conditions.
- We can rewrite as .
-
For :
- We need two numbers that multiply to and add up to .
- The factors and satisfy these conditions.
- We can rewrite as .
Thus, we have:
Step 2: Find the Least Common Denominator (LCD)
The LCD for the two fractions is the product of all unique factors in the denominators:
Step 3: Rewrite Each Fraction with the LCD
Now we express each fraction with the LCD as its denominator.
-
For :
- Multiply the numerator and denominator by to get:
-
For :
- Multiply the numerator and denominator by to get:
Step 4: Add the Fractions
Now we can combine the fractions since they have a common denominator:
Step 5: Simplify the Numerator
Expand and combine like terms in the numerator:
- Expanding :
- Expanding :
Adding these gives:
So, the expression becomes:
Step 6: Factor the Numerator if Possible
We can factor out a from the numerator:
Step 7: Simplify
Now, cancel the common factor of from the numerator and the denominator:
Final Answer
Would you like any further details on each step, or do you have any questions?
Here are some related questions that expand on this topic:
- How do you factor a quadratic expression?
- What is the process for finding the least common denominator?
- Why is it important to factor before adding rational expressions?
- How can you determine if further simplification is possible after adding fractions?
- How do you handle complex rational expressions with multiple terms in the numerator?
Tip: Always double-check factorizations to ensure accuracy when working with rational expressions.
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Math Problem Analysis
Mathematical Concepts
Algebra
Rational Expressions
Quadratic Equations
Formulas
Factoring quadratics
Least common denominator (LCD)
Theorems
Factoring theorem for quadratic expressions
Suitable Grade Level
Grades 10-12
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