Problem 27 - Algebra and Trigonometry

Question

Simplify the rational expression. $$ \frac{2 x^{2}-x^{2}-6 x}{2 x^{2}-7 x+6} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{x(x - 6)}{(2x - 3)(x - 2)} \).

Step 1: Simplify the Numerator

The given rational expression has a numerator of \( 2x^2 - x^2 - 6x \). First, combine like terms in the numerator: \( 2x^2 - x^2 = x^2 \). Thus, the simplified numerator becomes \( x^2 - 6x \).

Step 2: Simplify the Denominator

The denominator of the expression is \( 2x^2 - 7x + 6 \). Factor the quadratic expression. Find two numbers that multiply to \( 2 \times 6 = 12 \) and add to \(-7\). The numbers \(-3\) and \(-4\) satisfy this. Rewriting as \( 2x^2 - 4x - 3x + 6 \), group and factor: \( 2x(x - 2) - 3(x - 2) = (2x - 3)(x - 2) \).

Step 3: Factor the Numerator

Factor the simplified numerator \( x^2 - 6x \). Take \( x \) as a common factor: \( x(x - 6) \). The factored form of the numerator is \( x(x - 6) \).

Step 4: Simplify the Rational Expression

Substitute the simplified forms from Steps 1 and 2 into the original rational expression: \( \frac{x(x - 6)}{(2x - 3)(x - 2)} \). Check if there are any common factors in the numerator and the denominator. There are none to cancel, thus this is the simplified form.

Key Concepts

Factoring Quadratic Expressions

Factoring quadratic expressions is a crucial skill in simplifying rational expressions. A quadratic expression is typically in the form \( ax^2 + bx + c \). The key to factoring these expressions is finding two numbers that multiply to the product \( ac \) and add up to \( b \). This might sound complex, but with practice, it becomes intuitive.
To start:

• Identify \( a \), \( b \), and \( c \) in the equation.
• Calculate \( ac \), which means multiplying \( a \) and \( c \).
• Find two numbers that multiply to \( ac \) and sum to \( b \).

Once you have these numbers, you can rewrite the middle term and factor by grouping. In our step-by-step solution, we had \( 2x^2 - 7x + 6 \). After identifying the numbers \(-3\) and \(-4\), we grouped and factored to reach \((2x - 3)(x - 2)\). Practice with different quadratic expressions helps reinforce this method.

Common Factors in Algebra

Common factors in algebra refer to terms that appear in both the numerator and the denominator, which can be canceled to simplify an expression. It's like looking for shared traits and removing them. This is essential in simplifying rational expressions.
For any expression, the steps include:

• Identify any common factors in each term.
• Factor these out using parentheses.
• Cancel the common factors present in both the numerator and denominator if possible.

In our expression, the numerator after simplification was \( x(x - 6) \) and the denominator was \((2x - 3)(x - 2)\). Detection and cancelation of common factors simplify the rational expression. However, in this case, there were no common factors to cancel.

Combining Like Terms

Combining like terms is an initial step in simplifying algebraic expressions. Like terms share the same variable parts and can be added or subtracted. This process reduces the complexity of the expression and aids in further operations.
To combine like terms, simply:

• Identify terms with the same variable and power.
• Add or subtract the coefficients of these terms.

In the given solution, the numerator \( 2x^2 - x^2 - 6x \) contained like terms \( 2x^2 \) and \( -x^2 \). We combined these to get \( x^2 \), simplifying our work before factoring. This practice not only declutters expressions but also sets a foundation for further algebraic manipulations.