Problem 22 - Algebra and Trigonometry

Question

1–54 ? Find all real solutions of the equation. $$ \frac{x+\frac{2}{x}}{3+\frac{4}{x}}=5 x $$

Step-by-Step Solution

Answer

There are no real solutions for the equation.

Step 1: Understand the Equation

The given equation is \( \frac{x + \frac{2}{x}}{3 + \frac{4}{x}} = 5x \). This is a rational equation that involves both fractions and polynomial terms.

Step 2: Clear the Equation of Fractions

Multiply both sides of the equation by the denominator \(3 + \frac{4}{x}\) to eliminate the fraction on the left side. This gives us: \[ x + \frac{2}{x} = 5x(3 + \frac{4}{x}) \].

Step 3: Simplify the Right Side

Distribute \(5x\) on the right side of the equation. This results in: \[ x + \frac{2}{x} = 15x + \frac{20}{x} \].

Step 4: Move All Terms to One Side

Bring all terms to one side to set the equation to zero: \[ x + \frac{2}{x} - 15x - \frac{20}{x} = 0 \]. Combine like terms: \[ -14x + \frac{2}{x} - \frac{20}{x} = 0 \]. Simplify: \[ -14x - \frac{18}{x} = 0 \].

Step 5: Multiply by x to Clear the Fraction

To eliminate the fraction, multiply the entire equation by \(x\): \[ -14x^2 - 18 = 0 \].

Step 6: Solve for x

Rearrange the equation: \[ 14x^2 = -18 \]. Divide both sides by 14: \[ x^2 = -\frac{18}{14} \]. Simplify: \[ x^2 = -\frac{9}{7} \]. Since \(x^2\) cannot be negative in the real number system, no real solutions exist.

Key Concepts

Real Solutions

When we talk about real solutions of an equation, we refer to the values of the variable that satisfy the equation under the real number system. The real numbers include both positive and negative numbers, as well as zero. Importantly, they do not include imaginary or complex numbers.In the given problem, after simplifying the equation, we arrive at a point where the term is \(x^2 = -\frac{9}{7}\). Here, \(x^2\) is meant to represent the square of a number. In the realm of real numbers, a square cannot be negative. Thus, in this context, there are no real solutions to satisfy the equation because we cannot find a real number whose square is negative. This is why complex numbers, which can include the square root of negative numbers, are used in a different context. But for our real solutions focus, this equation has none.

Rational Equations

Rational equations, like the one in this exercise, are equations that contain at least one rational expression. A rational expression is a fraction in which both the numerator and the denominator are polynomials.Key points to remember about rational equations:

• The first step in solving these equations is to eliminate the fractions by multiplying through by the least common denominator.
• Clearing fractions simplifies the process and often transforms the equation into a more manageable polynomial form.
• It is crucial to check if any solutions might result in a zero denominator in the original equation, as these are not valid.

In our example, clearing fractions involved multiplying both sides by the denominator of the left side, \(3 + \frac{4}{x}\), to transform the equation.

Polynomial Terms

Polynomial terms are the building blocks of expressions and equations that involve whole number powers of a variable. In context to the exercise, polynomial terms appear both inside the fractional expressions and as standalone elements.For instance, \(-14x^2\) is a polynomial term where \(x\) is raised to the power of 2, denoting a quadratic term. These terms are crucial as they dictate the degree and behavior of the equation.When solving equations with polynomial terms, you:

• First ensure all terms are properly combined and simplified.
• Look to set the equation to zero on one side to enable analysis or solution finding.
• Use algebraic manipulations such as factoring, expanding, or using the quadratic formula when necessary.

Recognizing the polynomial aspects in an equation helps in effectively simplifying and solving it, especially when combined with other concepts like rational expressions. In this exercise, we handled polynomial terms aligned with rational expressions to untangle and attempt to solve the equation.