Math Problem Statement

Nilai x maksimum yang memenuhi: √x√x√x√x.... = √4x + √4x + √4x + √4x....

Solution

This problem involves an infinite nested square root expression. To find the maximum value of xx, we can set the expression equal to a variable yy, and work with that.

Given the equation:

xxxx=4x+4x+4x+4x+=y\sqrt{x \sqrt{x \sqrt{x \sqrt{x \dots}}}} = \sqrt{4x + \sqrt{4x + \sqrt{4x + \sqrt{4x + \dots}}}} = y

Step-by-Step Solution:

  1. Substitute the Nested Root: Since the square roots repeat infinitely, you can set the entire expression equal to yy:

    y=4x+yy = \sqrt{4x + y}

  2. Square Both Sides: To eliminate the square root, square both sides of the equation:

    y2=4x+yy^2 = 4x + y

  3. Rearrange the Equation: Now, solve for xx:

    y2y=4xy^2 - y = 4x x=y2y4x = \frac{y^2 - y}{4}

  4. Solve for xx when the expression is stable: At the stable value of the infinite nested square root, we want xx to be equal to the value inside the square root. Thus, substitute x=yx = y into the equation:

    y=y2y4y = \frac{y^2 - y}{4}

  5. Multiply by 4 to simplify:

    4y=y2y4y = y^2 - y y25y=0y^2 - 5y = 0

  6. Factor the Equation:

    y(y5)=0y(y - 5) = 0

  7. **

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Math Problem Analysis

Mathematical Concepts

Infinite Series
Nested Square Roots
Algebra

Formulas

y = √(4x + y)
y^2 - y = 4x
y(y - 5) = 0

Theorems

Nested Radical Theorem

Suitable Grade Level

Grades 10-12