Problem 5 - Elementary and Intermediate Algebra

Question

When we simplify \(\frac{x^{2}+5 x}{4 x+20},\) the result is \(\frac{x}{4} .\) These equivalent expressions have the same value for all real numbers, except \(x=-5 .\) Show that they have the same value for \(x=1\)

Step-by-Step Solution

Answer

Both expressions give \(\frac{1}{4}\) for \(x=1\), confirming their equivalence at this point.

Step 1: Factor the Numerator and Denominator

First, factor out the greatest common factor from both the numerator and the denominator of the expression \(\frac{x^2 + 5x}{4x + 20}\). The numerator \(x^2 + 5x\) can be factored as \(x(x + 5)\). The denominator \(4x + 20\) can be factored as \(4(x + 5)\).

Step 2: Cancel Common Factors

Since the numerator and denominator both have a common factor of \(x + 5\), we can cancel this term from the expression. This simplification leaves us with \(\frac{x}{4}\).

Step 3: Verify Simplified Expression at Specified Value

Evaluate both the original expression and the simplified expression at \(x = 1\). Substitute \(x = 1\) into the original expression: \(\frac{1^2 + 5 \times 1}{4 \times 1 + 20} = \frac{1 + 5}{4 + 20} = \frac{6}{24}\). Simplifying \(\frac{6}{24}\) results in \(\frac{1}{4}\).

Step 4: Check the Simplified Expression

Substitute \(x = 1\) into the simplified expression \(\frac{x}{4}\): \(\frac{1}{4}\).

Step 5: Confirm Equivalence

Both expressions yield \(\frac{1}{4}\) when \(x = 1\). Thus, they have the same value at this particular point.

Key Concepts

Factoring Expressions

Factoring an expression is like breaking it down into smaller pieces or parts that can be multiplied together to form the original expression. Imagine you have a big box (the expression), and you want to find out what smaller boxes can fit inside to recreate the big box. This is what factoring does.

For the expression \( \frac{x^2 + 5x}{4x + 20} \), we need to factor both the numerator and the denominator.

• For the numerator \( x^2 + 5x \), the greatest common factor (GCF) is \( x \). By factoring it, we get \( x(x + 5) \).
• For the denominator \( 4x + 20 \), the GCF is \( 4 \). By factoring it, we get \( 4(x + 5) \).

The act of finding these common factors and breaking them into products is crucial for the next steps in simplification. It allows us to simplify complicated expressions into their core components.

Equivalent Expressions

Two expressions are equivalent if they have the same value for all values of the variables involved, where these values are defined. This means no matter what number you substitute for the variable, the two expressions should get you the same result. Equivalent expressions are just different ways of writing the same mathematical truth.

In our exercise,

• The original expression is \( \frac{x^2 + 5x}{4x + 20} \).
• The simplified expression is \( \frac{x}{4} \).

Both these expressions will give the same result for any value of \( x \) except \( x = -5 \). This exclusion exists because at \( x = -5 \), the denominator of the original expression becomes zero, which is mathematically undefined. But everywhere else, these expressions are equivalent, as seen by both evaluating to \( \frac{1}{4} \) when \( x = 1 \). By understanding equivalent expressions, students can recognize that simplifying expressions doesn't change their inherent value.

Canceling Common Factors

Canceling common factors is a way to simplify fractions by removing parts of a fraction that appear in both the numerator and the denominator. Think of it like trimming excess parts that don't impact the expression's value.

In the case of \( \frac{x^2 + 5x}{4x + 20} \), after factoring:

• The numerator becomes \( x(x + 5) \).
• The denominator becomes \( 4(x + 5) \).

Both expressions have a common factor of \( (x + 5) \).

By canceling out \( (x + 5) \) from both the numerator and the denominator, we are left with \( \frac{x}{4} \). It's important to remember this process can't be used if the factor involves division by zero, which is why \( x = -5 \) is excluded. Canceling doesn't change the value of an expression, it simplifies the form while keeping it equivalent. Always check that the terms being canceled are truly factors across the numerator and denominator.