Problem 21 - Prealgebra and Introductory Algebra

Question

Simplify. $$\frac{x^{2}+3 x-10}{x^{2}+2 x-8}$$

Step-by-Step Solution

Answer

The simplified form of the provided expression is \(\frac{x+5}{x+4}\).

Step 1: Factor the numerator

The expression in the numerator is \(x^2+3x-10\), a quadratic expression which can be factored into \((x-2)(x+5)\) as \(x^2 -2x + 5x -10 = x^2+3x-10 \)

Step 2: Factor the denominator

The expression in the denominator is \(x^2+2x-8\), a quadratic expression which can be factored into \((x-2)(x+4)\) as \(x^2 -2x + 4x -8 = x^2+2x-8\)

Step 3: Cancel out common factors

Now, we have \(\frac{(x-2)(x+5)}{(x-2)(x+4)}\). The common factor of \((x-2)\) in the numerator and the denominator can be cancelled out. This gives us our simplified fraction, \(\frac{x+5}{x+4}\).

Key Concepts

Factoring quadratics

Factoring quadratics is a key skill in simplifying rational expressions. When you come across a quadratic expression, such as \(x^2 + 3x - 10\), the goal is to rewrite it as a product of two binomials. This involves finding two numbers that both add to the middle term (3 in this case) and multiply to the constant term (-10). For \(x^2 + 3x - 10\), the factors are \(-2\) and \(5\), because \(-2 + 5 = 3\) and \(-2 \times 5 = -10\).
Another example is the quadratic \(x^2 + 2x - 8\). This factors into \((x-2)(x+4)\) because \(-2 + 4 = 2\) and \(-2 \times 4 = -8\). By practicing this technique, you will become more comfortable picking the correct factors that break down a quadratic expression into simpler components.
Learning different methods for factoring, such as the "ac method" or simply recognizing patterns, can also be helpful as you encounter more complex quadratic expressions.

Canceling common factors

Canceling common factors in rational expressions is similar to simplifying fractions. Once you factor both the numerator and the denominator, check for common terms. In the given exercise, both the numerator \((x-2)(x+5)\) and denominator \((x-2)(x+4)\) have the common factor \((x-2)\).
By cancelling this common factor from both numerator and denominator, you're left with the simplified expression \(\frac{x+5}{x+4}\).

• "Cancel out" only works when the same factor is present in both the top and bottom.
• Ensure the factors appear in full; partial terms cannot be cancelled.
• Always factor first before attempting to cancel to reveal any hidden commonalities.

Cancelling common factors helps reduce complex rational expressions into simpler forms without changing their value. This simplification is crucial for ease of handling and further mathematical operations.

Quadratic expressions

Quadratic expressions are equations of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. These expressions produce parabolas when graphed and have significant importance in algebra.
They often appear in various forms of algebra problems, including rational expressions. Simplifying a rational expression involves understanding the quadratic parts, breaking them down, and managing them effectively.
You can rewrite quadratic expressions by factoring, which turns them into products of linear expressions. The expression \(x^2 + 3x - 10\), for example, becomes \((x-2)(x+5)\).
Recognizing which quadratics can be factored directly, and which require additional steps, is a handy skill. You might encounter quadratics that need methods like completing the square or using the quadratic formula for factorization.
Appreciating the nature of quadratic expressions makes it easier to manage and solve problems involving them, particularly those dealing with division and rational expressions.