Problem 2 - Algebra 2 and Trigonometry

Question

Bebe said that since each of the denominators in the complex fraction \(\frac{\frac{d}{4}+\frac{3}{5}}{2-\frac{d^{2}}{2}}\) is a non-zero constant, the fraction is defined for all values of \(d\). Do you agree with Bebe? Explain why or why not.

Step-by-Step Solution

Answer

Bebe is incorrect; the fraction is undefined at \(d = 2\) and \(d = -2\) because the entire denominator becomes zero.

Step 1: Rewrite the Complex Fraction

A complex fraction is one where the numerator, denominator, or both contain a fraction. Here, \[ \frac{\frac{d}{4} + \frac{3}{5}}{2 - \frac{d^2}{2}} \] represents such a fraction. To analyze it, express the complex fraction in terms that might help to identify when it is undefined.

Step 2: Identify Conditions for Denominator

For any fraction to be defined, its denominator must not be zero. So, let's focus on the denominator of the complex fraction, which is \[ 2 - \frac{d^2}{2} .\] Set it to zero and solve for d: \[ 2 - \frac{d^2}{2} = 0 \] leads to \[ 2 = \frac{d^2}{2} \]. Multiplying both sides by 2 gives \[ 4 = d^2. \] Thus, \( d \) could be \( 2 \) or \( -2 \).

Step 3: Evaluate the Denominator Values

This means that the denominator is zero when \( d = 2 \) or \( d = -2 \), which makes the fraction undefined at these points. Therefore, the fraction is not defined for all values of \( d \). These specific values cause the entire denominator of the complex fraction to become zero.

Step 4: Summarize the Solution to Address Bebe's Statement

Bebe's statement is incorrect because although the smaller denominators (4 and 5) in the fraction components are constants and non-zero, the denominator of the entire complex fraction itself, \(2-\frac{d^2}{2}\), becomes zero at \(d = 2\) and \(d = -2\). Thus, the complex fraction is not defined for these values of \(d\).

Key Concepts

Undefined Fractions

Fractions become undefined when their denominators equal zero. This is a fundamental concept in mathematics. When you divide by zero, the result does not exist in the real number system. This phenomenon is why some fractions are considered 'undefined'.
For any fraction, ensure the denominator is never zero. Check each variable or expression that represents the denominator. If there's a value that makes the denominator zero, the fraction is undefined at that point. In complex fractions, you need to analyze the main denominator carefully to see under what conditions, if any, it becomes zero.

Denominator Zero

Encountering a zero in the denominator is pivotal in determining whether a fraction is valid. For instance, in the complex fraction \( \frac{\frac{d}{4} + \frac{3}{5}}{2 - \frac{d^2}{2}} \), we need to focus on the larger denominator, \(2 - \frac{d^2}{2}\). If this expression equals zero, the fraction becomes undefined.

• Set \(2 - \frac{d^2}{2} = 0\)
• Solving gives \(d^2 = 4\)
• Thus, \(d = 2\) or \(d = -2\)

The denominator is exactly zero when \(d\) is \(2\) or \(-2\), which makes the fraction undefined for these values.

Solving Equations

Solving equations in fractions often involves finding special conditions where the equation does not hold. Consider the example where you set the denominator equal to zero to find critical points. This is not only crucial in proving if a fraction is undefined but also in verifying solutions.
Finding solutions:

• Isolate the expression resembling zero.
• Solve the equation by performing operations that keep the balance of the equation.
• Check the solutions and determine their applicability.

In our example, setting \(2 - \frac{d^2}{2} = 0\) reveals \(d^2 = 4\), leading us to important points \(d = 2\) and \(d = -2\). Ensure these don't make other parts of the expression invalid.

Algebraic Expressions

Algebraic expressions form the basis of complex fractions and can influence when they are defined or undefined. Manipulating these expressions involves simplifying, substituting values, and balancing equations to evaluate correctness.
When working with algebraic expressions in fractions:

• Identify and separate terms in the numerator and denominator.
• Simplify any common factors if possible.
• Analyze the complete expression for zero indicators.

In complex fractions like \(\frac{\frac{d}{4} + \frac{3}{5}}{2 - \frac{d^2}{2}}\), understanding each term's impact on the overall fraction is key to finding when it is undefined. Algebraic manipulation helps in making these determinations clear and concise.