Solve $\frac{x+4}{4 x}-\frac{x+4}{4}=\frac{x-1}{4}$.
A. $x=-2$ or $x=1$
B. $x=\frac{3 \pm \sqrt{17}}{2}$
C. $x=-5 / 2$ or $x=1$
D. $x=\frac{5 \pm \sqrt{65}}{4}$

Multiply both sides by 4 x and expand: ( x + 4 ) − x ( x + 4 ) = x ( x − 1 ) becomes x + 4 − x 2 − 4 x = x 2 − x .
Simplify and rearrange to standard quadratic form: 2 x 2 + 2 x − 4 = 0 , then x 2 + x − 2 = 0 .
Factor the quadratic equation: ( x + 2 ) ( x − 1 ) = 0 .
Solve for x : x = − 2 or x = 1 . The final answer is x = − 2 or x = 1 ​ .

Explanation

Problem Analysis We are given the equation 4 x x + 4 ​ − 4 x + 4 ​ = 4 x − 1 ​ and asked to solve for x . Our goal is to isolate x and find its possible values.

Eliminating Fractions and Expanding First, we multiply both sides of the equation by 4 x to eliminate the fraction in the first term. This gives us ( x + 4 ) − x ( x + 4 ) = x ( x − 1 ) Expanding the terms, we get x + 4 − x 2 − 4 x = x 2 − x Combining like terms on the left side, we have − x 2 − 3 x + 4 = x 2 − x

Rearranging to Quadratic Form Next, we move all terms to one side to set the equation to zero: 2 x 2 + 2 x − 4 = 0 We can divide the entire equation by 2 to simplify it: x 2 + x − 2 = 0

Factoring and Solving Now, we factor the quadratic equation: ( x + 2 ) ( x − 1 ) = 0 This gives us two possible solutions for x :
x + 2 = 0 ⇒ x = − 2 x − 1 = 0 ⇒ x = 1

Final Solutions Therefore, the solutions are x = − 2 and x = 1 .


Examples
Consider a scenario where you are distributing resources, and the equation represents a balance that needs to be achieved. Solving this equation helps determine the values of x that satisfy the balance condition. For example, x could represent the number of workers assigned to a task, and the equation ensures that the workload is evenly distributed. Understanding how to solve such equations is crucial in resource management and optimization.

Answered by GinnyAnswer | 2025-07-08