Problem 58 - Intermediate Algebra

Question

Simplify each complex fraction. $$ \frac{\frac{2}{t-2}-\frac{1}{t^{2}+t-6}}{\frac{4}{t+3}} $$

Step-by-Step Solution

Answer

The simplified complex fraction is \( \frac{2t + 5}{4(t-2)} \), with \( t \neq -3 \) to avoid division by zero.

Step 1: Analyze the Complex Fraction

The given expression is a complex fraction with a numerator and denominator that are themselves fractions. Let's look at the larger fraction: the numerator is \( \frac{2}{t-2} - \frac{1}{t^{2}+t-6} \), and the denominator is \( \frac{4}{t+3} \). Our goal is to simplify this expression step by step.

Step 2: Factor the Quadratic Expression in Denominator

The denominator in the numerator's second term is \( t^2 + t - 6 \). We need to factor this quadratic. The expression factors to \((t-2)(t+3)\). So, the complex fraction becomes:- Numerator: \( \frac{2}{t-2} - \frac{1}{(t-2)(t+3)} \)- Denominator: \( \frac{4}{t+3} \).

Step 3: Find a Common Denominator for the Numerator

The common denominator for the numerator \( \frac{2}{t-2} - \frac{1}{(t-2)(t+3)} \) is \((t-2)(t+3)\). Rewrite each term with this common denominator:- \( \frac{2}{t-2} = \frac{2(t+3)}{(t-2)(t+3)} \)- The numerator becomes: \( \frac{2(t+3) - 1}{(t-2)(t+3)} \).

Step 4: Simplify the Numerator

Expand and simplify the numerator:- \( 2(t+3) - 1 = 2t + 6 - 1 = 2t + 5 \).- Therefore, the numerator is simplified to \( \frac{2t + 5}{(t-2)(t+3)} \).

Step 5: Simplify the Overall Complex Fraction

Now divide the simplified numerator by the denominator:\[ \frac{\frac{2t + 5}{(t-2)(t+3)}}{\frac{4}{t+3}} \]To divide by a fraction, multiply by its reciprocal:\[ \frac{2t + 5}{(t-2)(t+3)} \times \frac{t+3}{4} \]This simplification leads to:\[ \frac{(2t + 5)(t+3)}{4(t-2)(t+3)} \].

Step 6: Cancel Common Factors

The \((t+3)\) terms cancel out from the numerator and the denominator, as long as \(t eq -3\):\[ \frac{2t + 5}{4(t-2)} \].This is the simplified form of the original complex fraction.

Key Concepts

Factoring Quadratics

Factoring quadratics is a fundamental skill in algebra. When dealing with quadratic expressions like \( t^2 + t - 6 \), the goal is to express it as a product of two binomials. This method is crucial when simplifying complex fractions involving quadratics.
To factor the quadratic \( t^2 + t - 6 \), we need to find two numbers that multiply to \(-6\) and add up to \(1\). These numbers are \(3\) and \(-2\). So, the quadratic factors into \((t - 2)(t + 3)\).

• The first term in each binomial comes from splitting \( t^2 \) into \( t \times t \).
• The signs depend on the middle term and the constant at the end.
• Always double-check by expanding the factors to verify the original quadratic.

Being able to switch between factored and expanded forms allows you to simplify parts of complex fractions effectively, as each format serves different needs in calculations.

Simplifying Fractions

Simplifying fractions is a key concept that involves reducing a fraction to its simplest form, which makes it easier to work with. Complex fractions feature fractions within fractions and often involve breaking them down step by step.
Consider the numerator of our original expression, \( \frac{2}{t-2} - \frac{1}{(t-2)(t+3)} \), which needs simplification. After factoring, find a common denominator, which merges separate fractions into a single fraction that can be simplified.

• Rewrite each term using the common denominator.
• Combine the numerators into a single expression.
• Simplify by combining like terms and reducing if possible.

Remember, simplifying fractions is crucial when preparing to multiply or divide fractions, as seen with the reciprocal step in the problem. Each reduction step saves effort in later calculations.

Finding Common Denominators

Finding common denominators is vital when you're working with multiple fractions, such as in complex fractions. A common denominator enables the combination of fractions by aligning them under one single denominator.
In the numerator \( \frac{2}{t-2} - \frac{1}{(t-2)(t+3)} \), the common denominator is \((t-2)(t+3)\). This choice is strategic because the resulting expression can be used uniformly with all terms.

• Transform each fraction so they share the same denominator.
• This might involve multiplying the numerator and denominator of one or more fractions by missing components.
• Once aligned, it becomes straightforward to combine or subtract the fractions.

When you have the common denominators in place, it clarifies how separate parts fit into a unified whole and prepares for the next simplification steps. Being proficient in finding common denominators simplifies the complex fraction process significantly.