Problem 14 - A Graphical Approach to Precalculus with Limits

Question

Find all complex solutions for each equation by hand. $$\frac{8 x}{4 x^{2}-1}=\frac{3}{2 x+1}+\frac{3}{2 x-1}$$

Step-by-Step Solution

Answer

The complex solution is \(x = 0\).

Step 1: Combine the Fractions on the Right

Combine the fractions on the right-hand side by finding a common denominator. The common denominator for \(2x+1\) and \(2x-1\) is \((2x+1)(2x-1)\). Thus,\[\frac{3}{2x+1} + \frac{3}{2x-1} = \frac{3(2x-1) + 3(2x+1)}{(2x+1)(2x-1)} = \frac{6x}{4x^2-1}.\]

Step 2: Simplify and Set the Equation

Substitute the simplified fraction back into the equation and equate the left-hand side and the right-hand side:\[\frac{8x}{4x^2-1} = \frac{6x}{4x^2-1}.\]

Step 3: Equate Numerators

Since the denominators are the same, equate the numerators directly. This gives:\[8x = 6x.\]

Step 4: Solve the Simplified Equation

Solve the equation obtained by equating the numerators:\[8x - 6x = 0 \Rightarrow 2x = 0 \Rightarrow x = 0.\]

Step 5: Check the Validity of the Solution in the Original Equation

Substitute \(x = 0\) back into the original equation to ensure it doesn't make any denominator zero.- For the left-hand side: \(4(0)^2 - 1 = -1\Rightarrow \text{denominator is valid.}\)- For the right-hand side: \((2(0)+1)(2(0)-1)=-1\Rightarrow \text{valid as well.}\)Since \(x = 0\) does not make any denominator zero, it is indeed a valid solution.

Key Concepts

Equation Solving

Solving equations is a fundamental skill in mathematics. It involves finding the values of variables that make an equation true. Some equations are straightforward, while others can be quite complex. In general, here are some tips when solving equations:

• Identify the type of equation: Understand whether you're dealing with a linear, quadratic, or another type of equation.
• Manipulate the equation: Use operations like addition, subtraction, multiplication, or division to simplify or rearrange the equation.
• Isolate the variable: Your goal is often to get the variable on one side of the equation to find its value.
• Check your solution: Substitute the value back into the original equation to verify its correctness.

In more complex problems, like the one described, it might involve combining fractions or setting parts of the equation equal after simplification.

Complex Solutions

Complex solutions come into play when dealing with quadratic or higher polynomial equations. These solutions include complex numbers, which have real and imaginary parts. A complex number is expressed in the form of:

• \( a + bi \) where \( a \) is the real part and \( b \) is the imaginary coefficient, and \( i \) is the square root of -1.

In some cases, when solving equations, you may encounter situations where the solution indicates taking the square root of a negative number. This is when complex numbers serve as valid solutions.
To incorporate complex numbers:

• Recognize the discriminant: If a quadratic equation has a negative discriminant, it means the equation has complex solutions.
• Use the quadratic formula: To find these solutions, use the quadratic formula, which inherently accounts for complex solutions.

Though the given problem yields a real solution, understanding complex solutions is crucial for a comprehensive grasp of solving equations.

Fractions

Working with fractions, especially in equations, can be challenging. Mastering fractions involves understanding how to add, subtract, multiply, and divide them. When solving equations with fractions, remember:

• Find a common denominator: This step is critical when adding or subtracting fractions. It makes the fractions easier to work with.
• Simplify the fractions: After combining fractions, reduce them to their simplest form for easier calculation.
• Cross-multiply: When fractions are set equal to each other, cross-multiplying can help eliminate the fractional terms.

In the provided exercise, the fractions on the right-hand side of the equation were combined by finding a common denominator. This allowed them to be integrated into the equation straightforwardly. Understanding fractions is essential not only for current problems but also for future mathematical concepts.