Problem 42 - Algebra 1: Concepts and Skills

Question

Simplify the expression if possible. $$ \frac{3 x-5}{25-30 x+9 x^{2}} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(\frac{1}{3x - 5}\).

Step 1: Reordering

Firstly, reorder the denominator of the given expression \(\frac{3x-5}{25 - 30x + 9x^2}\) to get \(\frac{3x-5}{9x^2 - 30x + 25}\). This makes it easier to spot any potential factorizations.

Step 2: Factor

Now, factor the numerator and the denominator. The numerator can't be factored further, but the denominator can. The denominator is a quadratic expression, which follows the form \(ax^2 + bx + c\), where a = 9, b = -30, and c = 25. It can be factored into \((3x - 5)(3x - 5)\) or \((3x - 5)^2\). So, the expression can be rewritten as \(\frac{3x-5}{(3x - 5)^2}\).

Step 3: Simplify

Seeing that \(3x - 5\) is a common term in the numerator and the denominator, it can be factored out and cancelled. This simplifies the original expression to \(\frac{1}{3x - 5}\).

Key Concepts

Factoring Quadratic Expressions

Factoring quadratic expressions is a fundamental skill in algebra that helps to simplify and solve equations. A quadratic expression typically looks like this: \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. The main goal is to rewrite this expression as a product of two binomials.

To factor a quadratic expression, you need to:

• Identify the coefficients \( a \), \( b \), and \( c \).
• Make sure the quadratic is in the standard form \( ax^2 + bx + c \).
• Find two numbers that multiply to \( ac \) (the product of the first and last coefficients) and add up to \( b \).
• Use these numbers to break up the middle term \( bx \).
• Factor by grouping if necessary.

In our exercise, the quadratic \( 9x^2 - 30x + 25 \) was factored into \((3x - 5)(3x - 5)\). This reveals a special form known as a perfect square trinomial. Understanding these forms helps you recognize patterns and simplify expressions more efficiently.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are both polynomials. The simplification of rational expressions involves making the expression as simple as possible while keeping the same value.

For rational expressions:

• Identify common factors in the numerator and the denominator.
• Factor both the numerator and the denominator fully if possible.
• Cancel any common factors that appear in both the numerator and the denominator.

In the given exercise, we have the rational expression \( \frac{3x-5}{9x^2 - 30x + 25} \). By factoring the denominator and noticing that \( 3x - 5 \) is present in both the numerator and denominator, we can simplify the expression. This is key to handling rational expressions, as the simplification often reveals clearer or more meaningful results.

Simplification Steps

Simplification of expressions is a process that reduces the expression to its most basic form, making it easier to interpret or solve. It follows a systematic order to ensure accuracy.

To simplify an expression:

• Reorder terms if necessary for improved clarity or factorization.
• Fully factor numerators and denominators.
• Cancel common factors between the numerator and denominator.
• Rewrite the expression in its simplest form.

In the exercise, after reordering and factoring, our expression becomes \( \frac{3x-5}{(3x - 5)^2} \). Since \( 3x - 5 \) is in both the numerator and denominator, they cancel to simplify the expression to \( \frac{1}{3x - 5} \). Simplification not only makes the expression neater but also helps identify any restrictions or domain considerations, such as when a denominator should not be zero.