Problem 57 - Elementary Algebra

Question

For the following problems, simplify each expressions. $$ \frac{\sqrt{x^{2}-4 x+3}}{\sqrt{x-3}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $$\frac{\sqrt{x^2 - 4x + 3}}{\sqrt{x - 3}}$$ Short Answer: The simplified expression is $$\sqrt{x - 1}$$.

Step 1: Factor the numerator

First, we want to determine if the numerator can be factored. Let's examine $$x^2 - 4x + 3$$. To factor it, we need to find two numbers that when multiplied, equal the last term (3) and when added, equal the middle term (-4). In this case, those numbers are -3 and -1. We can now rewrite the expression as: $$ \frac{\sqrt{(x-3)(x-1)}}{\sqrt{x-3}} $$

Step 2: Simplify the fraction

Now that we have factored the numerator, we can simplify the fraction by canceling out the common factors of the numerator and denominator. In this case, that factor is $$\sqrt{x - 3}$$. After canceling, we're left with: $$ \sqrt{x - 1} $$

Step 3: Final expression

The given expression has been fully simplified, and the final expression is: $$ \sqrt{x - 1} $$

Key Concepts

Factoring Quadratics

Understanding how to factor quadratics is essential for simplifying complex algebraic expressions. Quadratics are polynomial expressions of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. The goal is to break them down into simpler factors that, when multiplied, give us the original quadratic.

When factoring a quadratic, we look for two binomials that when expanded, would produce the original terms. Let's consider \( x^2 - 4x + 3 \). To factor it, we search for two numbers that multiply to the constant term, \( +3 \), and add up to the coefficient of the linear term, \( -4 \). In this example, these numbers are \( -3 \) and \( -1 \), yielding the factors \( (x - 3) \) and \( (x - 1) \). Therefore, \( x^2 - 4x + 3 \) can be expressed as \( (x - 3)(x - 1) \).

This process is a cornerstone of algebra because it enables the simplification of more complex expressions, such as those within radical signs or algebraic fractions.

Radical Simplification

Radical simplification involves the process of reducing expressions within a radical sign to their simplest form. A radical, often represented as the square root sign \( \sqrt{\phantom{x}} \), signifies the operation of finding a number that, when multiplied by itself, would yield the expression within the radical.

To simplify a radical expression such as \( \frac{\sqrt{x^2 - 4x + 3}}{\sqrt{x - 3}} \), we initially focus on simplifying the numerator by employing factoring techniques. By acknowledging that \( x^2 - 4x + 3 \) can be rewritten as \( (x-3)(x-1) \), we're one step closer to a simpler form. Since the denominator already contains a factor of \( x-3 \) within its radical, we can then cancel out the common factor between the numerator and the denominator. This cancellation process leads to the simplified expression \( \sqrt{x - 1} \), which no longer has a radical in the denominator.

Thus, understanding radical simplification is crucial in moving from complex root expressions to ones that are more tractable and easier to work with.

Algebraic Fractions

Algebraic fractions, akin to numerical fractions, consist of a numerator and a denominator. However, they can encompass literals (variables), constants, and various algebraic expressions. Simplifying algebraic fractions is key to solving many algebra problems. This process often involves factoring polynomials in the numerator and denominator and then reducing by canceling out common factors.

Consider the expression \( \frac{\sqrt{x^2-4x+3}}{\sqrt{x-3}} \). Here, we have a radical expression in both the numerator and the denominator, creating a complex algebraic fraction. Through factoring and radical simplification, as demonstrated in the above sections, we can convert \