(2x-3-5/x)(14/(x+1)+2+(sqrt(-1-2x))^2) =0

4 min read Jun 16, 2024

Solving the Equation: (2x-3-5/x)(14/(x+1)+2+(sqrt(-1-2x))^2) = 0

This equation presents a fascinating challenge due to its complexity. Let's break down the steps involved in finding its solutions:

1. Understanding the Structure

The equation is a product of two factors:

Factor 1: (2x - 3 - 5/x)
Factor 2: (14/(x+1) + 2 + (sqrt(-1-2x))^2)

For the product of two factors to equal zero, at least one of the factors must be zero.

2. Solving for Factor 1

2x - 3 - 5/x = 0

To simplify, multiply both sides by 'x':

2x² - 3x - 5 = 0

This is a quadratic equation that can be solved using the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

Where a = 2, b = -3, and c = -5

Plugging in the values:

x = (3 ± √((-3)² - 4 * 2 * -5)) / (2 * 2)

x = (3 ± √(49)) / 4

x = (3 ± 7) / 4

Therefore, the solutions for Factor 1 are:

x = 5/2
x = -1

3. Solving for Factor 2

14/(x+1) + 2 + (sqrt(-1-2x))^2 = 0

Simplify the square root:

14/(x+1) + 2 + (-1-2x) = 0

Combine terms:

14/(x+1) - 2x + 1 = 0

Multiply both sides by (x+1):

14 - 2x(x+1) + (x+1) = 0

Expand and rearrange:

-2x² - x + 15 = 0

Solve using the quadratic formula (a = -2, b = -1, c = 15):

x = (1 ± √((-1)² - 4 * -2 * 15)) / (2 * -2)

x = (1 ± √(121)) / -4

x = (1 ± 11) / -4

Therefore, the solutions for Factor 2 are:

x = -3
x = -5/2

4. Final Solutions

Combining the solutions from both factors, the complete set of solutions for the equation (2x-3-5/x)(14/(x+1)+2+(sqrt(-1-2x))^2) = 0 are:

x = 5/2
x = -1
x = -3
x = -5/2

It's important to note that we need to verify these solutions by plugging them back into the original equation to ensure they don't result in undefined expressions (like division by zero). In this case, x = -1 would lead to an undefined expression in Factor 2.

Therefore, the final set of valid solutions for the given equation are:

x = 5/2
x = -3
x = -5/2