Problem 25 - Algebra 1: Concepts and Skills

Question

FACTORING AFTER ADDING OR SUBTRACTING. Simplify the expression. $$ \frac{a^{2}-2}{a^{2}-25}+\frac{4 a-3}{a^{2}-25} $$

Step-by-Step Solution

Answer

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

Step 1: Recognise the common denominator

The given problem consists of two fractions. The denominator of both fractions is \(a^{2}-25\). This is a quadratic expression, which we can factor into \((a-5)(a+5)\). As the denominator for both fractions is the same, we can combine the two fractions by adding the numerators.

Step 2: Combine the fractions

When we add the two fractions, the new numerator becomes \((a^{2}-2) + (4 a-3)\). Therefore, the combined fraction is \(\frac{(a^{2}-2) + (4 a-3)}{(a-5)(a+5)}\) or \(\frac{a^{2}+4 a-2-3}{(a-5)(a+5)}\), which simplifies to \(\frac{a^{2}+4 a-5}{(a-5)(a+5)}\).

Step 3: Factor the resulting numerator

The numerator \(a^{2}+4 a-5\) of the combined fraction can be factored into \((a-1)(a+5)\). Hence, the algebraic expression simplifies to \(\frac{(a-1)(a+5)}{(a-5)(a+5)}\).

Step 4: Simplify the expression

We can now cancel out the common factors between the numerator and the denominator. In this case, (a+5) is common and can be cancelled out which will simplify the expression to \(\frac{a-1}{a-5}\).

Key Concepts

Quadratic Expressions

Quadratic expressions are a key concept in algebra, often appearing in the form \(ax^2 + bx + c\). In our exercise, the denominator \(a^2 - 25\) is a quadratic expression. This can be factored using the difference of squares formula into \((a-5)(a+5)\).
Understanding how to factor quadratic expressions helps us simplify and manipulate algebraic fractions. Factoring allows us to see the roots of the expressions and makes it easier to combine or reduce fractions later on.
Key points to remember:

• Identify if the expression is a perfect square or linked by addition/subtraction signs.
• Use formulas like difference of squares: \(a^2 - b^2 = (a-b)(a+b)\).
• Factoring helps simplify further calculations and find common factors in fractions.

Common Denominators

When working with fractions, recognizing a common denominator allows us to add or subtract them. In the given problem, both fractions share the denominator \(a^2 - 25\).
Factoring it into \((a-5)(a+5)\) not only simplifies the equation but also sets us up for adding the numerators.
Why is a common denominator important?

• It unites fractions so they can be combined into a single expression.
• Pre-factoring the denominator makes the process of addition/subtraction smooth.
• It leads to simpler results that are easier to interpret or further manipulate.

Finding and factoring a common denominator is crucial, especially in algebraic contexts. It reduces possible errors and clarifies complex expressions.

Simplifying Fractions

Simplifying fractions involves canceling common factors in the numerator and denominator. After combining the fractions, our numerator \(a^2 + 4a - 5\) was factored into \((a-1)(a+5)\).
The matching term \((a+5)\) in the denominator allows us to cancel and simplify the fraction.
Steps to simplify:

• Factor both numerator and denominator when possible.
• Identify and cancel any common factors.
• Ensure that the resulting expression is still valid under the domain constraints.

This process not only makes fractions easier to understand but also prepares them for further operations, like solving equations or integrating into more complex algebraic expressions.