Problem 90 - Algebra for College Students

Question

Perform the indicated operations. $$ \frac{2 x+1}{6 x^{2}-5 x+1}+\frac{2 x-1}{6 x^{2}+x-1} $$

Step-by-Step Solution

Answer

\(\frac{8x^2 + 2}{(2x - 1)(3x - 1)(2x + 1)}\)

Step 1: Simplify the Denominators

First, factorize both denominators. For the first fraction, we factorize the quadratic equation: \[6x^2 - 5x + 1 = (2x - 1)(3x - 1) \]For the second fraction, we factorize the quadratic equation: \[6x^2 + x - 1 = (2x + 1)(3x - 1)\]

Step 2: Rewrite the Fractions

Rewrite each fraction using the factored denominators: \[\frac{2x + 1}{(2x - 1)(3x - 1)} + \frac{2x - 1}{(2x + 1)(3x - 1)} \]

Step 3: Find the Common Denominator

The common denominator of the two fractions is \[(2x - 1)(3x - 1)(2x + 1) \]

Step 4: Adjust Numerators

Adjust the numerators so that both fractions have the common denominator: \[\frac{(2x + 1)(2x + 1)}{(2x - 1)(3x - 1)(2x + 1)} + \frac{(2x - 1)(2x - 1)}{(2x + 1)(3x - 1)(2x - 1)} \]Now simplify: \[\frac{(2x + 1)^2}{(2x - 1)(3x - 1)(2x + 1)} + \frac{(2x - 1)^2}{(2x + 1)(3x - 1)(2x - 1)} \]

Step 5: Combine the Fractions

Combine the fractions over the common denominator: \[\frac{(2x + 1)^2 + (2x - 1)^2}{(2x - 1)(3x - 1)(2x + 1)} \]

Step 6: Expand and Simplify the Numerator

Expand the numerators: \[(2x + 1)^2 = 4x^2 + 4x + 1 \]\[(2x - 1)^2 = 4x^2 - 4x + 1 \]Thus, \[(2x + 1)^2 + (2x - 1)^2 = 4x^2 + 4x + 1 + 4x^2 - 4x + 1 = 8x^2 + 2\]

Step 7: Final Fraction

Write the final simplified fraction: \[\frac{8x^2 + 2}{(2x - 1)(3x - 1)(2x + 1)} \]

Key Concepts

Factoring Quadratics

When working with algebraic fractions, it's crucial to recognize and factor quadratic expressions. This simplifies the expression and makes subsequent steps easier.
For example, consider the quadratic equation in the first fraction's denominator:
To factorize Finally, for the second fraction: This splitting of the quadratic equations into linear factors helps in finding a common denominator and simplifies addition or subtraction of fractions.

Common Denominator

Finding a common denominator is essential when adding or subtracting fractions. This is similar to the process used for simple numerical fractions, but here it involves polynomial expressions.
To find the common denominator of the fractions:
Factorize both denominators: Now, observe that both fractions share a common factor, multiplying all distinct factors gives the common denominator which is the product of all factors in both denominators.
This common denominator allows us to rewrite and combine the fractions effectively.

Simplifying Expressions

Simplifying expressions often involves several steps: rewriting, combining, expanding, and canceling terms where possible.
Once both fractions have a common denominator, the next steps include:

• Rewriting the numerators so that each fraction has the common denominator
• Combining the fractions into a single fractional expression

Expanding and simplifying the result helps in getting the final simplified expression:
1 and (2x - 1)
Combining and simplifying the numerators: leads to
Our final simplified fraction is: which is easier to interpret and use in further calculations.Understanding these simplification techniques helps in managing complex algebraic fractions more effectively.