Math Problem Statement

Add \(\frac{9}{4x^2 - 7x - 2} + \frac{3}{4x^2 + 5x + 1}\) and simplify your answer as much as possible.

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:

94x27x2+34x2+5x+1\frac{9}{4x^2 - 7x - 2} + \frac{3}{4x^2 + 5x + 1}

Step 1: Factor the Denominators

We start by factoring each quadratic expression in the denominator if possible.

  1. For 4x27x24x^2 - 7x - 2:

    • We need two numbers that multiply to 4×2=84 \times -2 = -8 and add up to 7-7.
    • The factors 8-8 and +1+1 satisfy these conditions.
    • We can rewrite 4x27x24x^2 - 7x - 2 as (4x+1)(x2)(4x + 1)(x - 2).
  2. For 4x2+5x+14x^2 + 5x + 1:

    • We need two numbers that multiply to 4×1=44 \times 1 = 4 and add up to 55.
    • The factors 44 and 11 satisfy these conditions.
    • We can rewrite 4x2+5x+14x^2 + 5x + 1 as (4x+1)(x+1)(4x + 1)(x + 1).

Thus, we have: 9(4x+1)(x2)+3(4x+1)(x+1)\frac{9}{(4x + 1)(x - 2)} + \frac{3}{(4x + 1)(x + 1)}

Step 2: Find the Least Common Denominator (LCD)

The LCD for the two fractions is the product of all unique factors in the denominators: (4x+1)(x2)(x+1)(4x + 1)(x - 2)(x + 1)

Step 3: Rewrite Each Fraction with the LCD

Now we express each fraction with the LCD as its denominator.

  1. For 9(4x+1)(x2)\frac{9}{(4x + 1)(x - 2)}:

    • Multiply the numerator and denominator by (x+1)(x + 1) to get: 9(x+1)(4x+1)(x2)(x+1)\frac{9(x + 1)}{(4x + 1)(x - 2)(x + 1)}
  2. For 3(4x+1)(x+1)\frac{3}{(4x + 1)(x + 1)}:

    • Multiply the numerator and denominator by (x2)(x - 2) to get: 3(x2)(4x+1)(x2)(x+1)\frac{3(x - 2)}{(4x + 1)(x - 2)(x + 1)}

Step 4: Add the Fractions

Now we can combine the fractions since they have a common denominator:

9(x+1)+3(x2)(4x+1)(x2)(x+1)\frac{9(x + 1) + 3(x - 2)}{(4x + 1)(x - 2)(x + 1)}

Step 5: Simplify the Numerator

Expand and combine like terms in the numerator:

  1. Expanding 9(x+1)9(x + 1): 9x+99x + 9
  2. Expanding 3(x2)3(x - 2): 3x63x - 6

Adding these gives: 9x+9+3x6=12x+39x + 9 + 3x - 6 = 12x + 3

So, the expression becomes:

12x+3(4x+1)(x2)(x+1)\frac{12x + 3}{(4x + 1)(x - 2)(x + 1)}

Step 6: Factor the Numerator if Possible

We can factor out a 33 from the numerator:

3(4x+1)(4x+1)(x2)(x+1)\frac{3(4x + 1)}{(4x + 1)(x - 2)(x + 1)}

Step 7: Simplify

Now, cancel the common factor of 4x+14x + 1 from the numerator and the denominator:

3(x2)(x+1)\frac{3}{(x - 2)(x + 1)}

Final Answer

3(x2)(x+1)\frac{3}{(x - 2)(x + 1)}

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:

  1. How do you factor a quadratic expression?
  2. What is the process for finding the least common denominator?
  3. Why is it important to factor before adding rational expressions?
  4. How can you determine if further simplification is possible after adding fractions?
  5. 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