Problem 45 - Introductory Algebra for College Students

Question

Solve each equation by the method of your choice. Simplify irrational solutions, if possible $$\frac{x^{2}}{x+7}-\frac{3}{x+7}=0$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x= \sqrt{3}\) and \(x= -\sqrt{3}\).

Step 1: Simplify the equation

To start, since both terms have the same denominator, i.e., \(x+7\), combine the terms over this common denominator. The equation therefore simplifies to \[\frac{x^{2}-3}{x+7}=0\].

Step 2: Set the numerator equal to zero

Since the denominator cannot be zero, it implies that the numerator is equal to zero; i.e., \(x^{2}-3=0\) which simplifies to \(x^{2}=3\).

Step 3: Solve for x

To solve this, take the square root of both sides, keeping in mind that the square root could be either positive or negative. The equation simplifies to \(x= \pm\sqrt{3}\).

Step 4: Check the validity of solutions

We can check if \(x= \sqrt{3}\) or \(x= -\sqrt{3}\) makes our original denominator equal to zero. Clearly, neither value makes \(x+7=0\). Therefore, both are valid solutions.

Key Concepts

Rational Expressions

Rational expressions are fractions that involve polynomials in their numerators and denominators. They play a key role in algebra as they help you understand division involving variables. A typical rational expression looks like \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) \) is not zero (since division by zero is undefined).
Understanding how to work with rational expressions involves:

• Identifying common denominators to combine or compare fractions.
• Simplifying the expressions by factoring polynomials and reducing the fraction.
• Ensuring the denominator never becomes zero, which is a critical restriction.

In our example, \( \frac{x^{2}}{x+7} - \frac{3}{x+7} = 0 \), we see both terms share a common denominator, \( x+7 \). This allows us to combine them easily into a single rational expression: \( \frac{x^{2} - 3}{x+7} = 0 \). When solving equations involving rational expressions, the focus is often on addressing the numerator once the equation is simplified, as the common denominator means we are setting the entire expression to zero.

Quadratic Equations

Quadratic equations take the form \( ax^2 + bx + c = 0 \) and are a fundamental concept in algebra. Solving these equations often involves techniques like factoring, completing the square, or using the quadratic formula.
In our simplified problem, the equation transforms into \( x^2 - 3 = 0 \), a basic quadratic form with \( a = 1, b = 0, \) and \( c = -3 \). Solving simple quadratic equations like this often involves:

• Isolating the \( x^2 \) term.
• Taking the square root of both sides, which results in two solutions: one positive and one negative.

Thus, \( x^2 = 3 \) simplifies further to \( x = \pm\sqrt{3} \). Always remember the "plus-minus" in front of the square root, as it ensures all potential solutions are considered.

Simplifying Expressions

Simplifying expressions involves reducing a given algebraic expression to its most concise form without changing its value. For rational expressions, this often means manipulating the numerator and the denominator to cancel out common factors.
In our exercise, we started with an expression \( \frac{x^{2}}{x+7} - \frac{3}{x+7} \). Simplification led us to \( \frac{x^{2}-3}{x+7} \), combining like terms over a common denominator.
Here are some key strategies for simplifying:

• Identify and factor common terms where possible to streamline your expression.
• Combine fractions by finding a common denominator when necessary.
• Check solutions by substituting them back to ensure they don't invalidate restrictions, like causing division by zero.

Simplification prepares an equation for further solving and reduces it to a form that can be easily analyzed and interpreted.