Problem 48 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ \frac{x^{2}+x-12}{2 x^{2}-9 x-5} \div \frac{x^{2}+7 x+12}{2 x^{2}-7 x-4} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{(x-3)(x-4)}{(x-5)(x+3)} \).

Step 1: Identify the Main Operation

The given expression involves a division between two fractions: \( \frac{x^{2}+x-12}{2x^{2}-9x-5} \) and \( \frac{x^{2}+7x+12}{2x^{2}-7x-4} \). The main operation is division.

Step 2: Rewrite Division as Multiplication

When dividing fractions, you multiply by the reciprocal of the divisor. So, rewrite the expression as \[ \frac{x^{2}+x-12}{2x^{2}-9x-5} \times \frac{2x^{2}-7x-4}{x^{2}+7x+12} \].

Step 3: Factor all Quadratic Expressions

Factor each quadratic in the numerators and denominators:- \(x^{2}+x-12 = (x+4)(x-3)\)- \(2x^{2}-9x-5 = (2x+1)(x-5)\)- \(x^{2}+7x+12 = (x+3)(x+4)\)- \(2x^{2}-7x-4 = (2x+1)(x-4)\)

Step 4: Substitute Factored Forms

Replace each quadratic expression in the original fraction with its factored form:\[ \frac{(x+4)(x-3)}{(2x+1)(x-5)} \times \frac{(2x+1)(x-4)}{(x+3)(x+4)} \].

Step 5: Simplify by Cancelling Common Terms

Cancel out the common terms from the numerator and the denominator:- \((x+4)\) appears in both a numerator and a denominator and can be cancelled.- \((2x+1)\) appears in both a numerator and a denominator and can be cancelled.The expression is now:\[ \frac{(x-3)(x-4)}{(x-5)(x+3)} \].

Step 6: Final Simplified Expression

The expression \( \frac{(x-3)(x-4)}{(x-5)(x+3)} \) is fully simplified, as there are no more common factors between the numerator and denominator.

Key Concepts

Quadratic Expressions

Quadratic expressions are a type of algebraic expression where the variable is raised to the power of two. In simpler terms, these expressions form a curve called a parabola when plotted on a graph. They usually come in the form of \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants.

In our exercise, we encounter several quadratic expressions such as \( x^2 + x - 12 \) and \( 2x^2 - 9x - 5 \). Understanding these expressions is crucial because they can represent real-world phenomena, like the trajectory of an object thrown in the air.

This form of expression must often be manipulated or rewritten to solve equations and simplify problems. For instance, factoring is a common method that makes quadratics easier to handle, as we will see in the next steps.

Factoring

Factoring is a key algebraic process that breaks down expressions into their simplest components called factors. It's like finding the basic building blocks of a number, only we are doing this with algebraic expressions.

In this case, factoring is applied to quadratic expressions to convert them from the standard form \( ax^2 + bx + c \) into a product of two binomials. Each of these binomials can be considered a relatively simpler expression.

For instance, the expression \( x^2 + x - 12 \) can be factored into \((x + 4)(x - 3)\). Similarly, \( 2x^2 - 9x - 5 \) becomes \((2x + 1)(x - 5)\) after factoring.

• Factoring reveals roots of the quadratic equation.
• Helps simplify complex expressions into more manageable forms.
• Allows elimination of similar terms in expressions to simplify them further.

Factoring is an invaluable tool for solving quadratic equations and simplifying expressions, making algebra more approachable.

Simplifying Expressions

Simplifying expressions means reducing them to their most basic form, making them easier to understand or solve. This involves eliminating redundant parts by canceling common factors or combining like terms.

In our exercise, after rewriting and factoring the quadratic expressions, simplifying involves finding terms that appear in both the numerator and the denominator and canceling them out. This step transforms \( \frac{(x+4)(x-3)}{(2x+1)(x-5)} \times \frac{(2x+1)(x-4)}{(x+3)(x+4)} \) into \( \frac{(x-3)(x-4)}{(x-5)(x+3)} \).

Why is simplifying important?

• It makes expressions easier to work with and solutions more accessible.
• Helps in identifying equivalent expressions.
• Can reveal more direct solutions to equations by removing complexity.

When you simplify an expression, you basically streamline the problem-solving process, which is essential in both academic and real-world applications.