The given equation is:

\[
\sqrt{5 \cdot x^2} - 4 \cdot x + 4 = 2 \cdot x
\]

Let's simplify and solve for \(x\).

### Step 1: Simplify the square root term
\[
\sqrt{5 \cdot x^2} = \sqrt{5} \cdot |x|
\]
Thus, the equation becomes:
\[
\sqrt{5} \cdot |x| - 4x + 4 = 2x
\]

### Step 2: Rearrange the terms
Move all terms to one side of the equation:
\[
\sqrt{5} \cdot |x| - 4x - 2x + 4 = 0
\]
This simplifies to:
\[
\sqrt{5} \cdot |x| - 6x + 4 = 0
\]

Now we have a piecewise case for \(|x|\), based on whether \(x\) is positive or negative.

### Step 3: Consider the cases for \(x\)

#### Case 1: \(x \geq 0\) (so \(|x| = x\))
The equation becomes:
\[
\sqrt{5}x - 6x + 4 = 0
\]
Factor out \(x\):
\[
(\sqrt{5} - 6)x + 4 = 0
\]
Solve for \(x\):
\[
(\sqrt{5} - 6)x = -4
\]
\[
x = \frac{-4}{\sqrt{5} - 6}
\]

#### Case 2: \(x < 0\) (so \(|x| = -x\))
The equation becomes:
\[
-\sqrt{5}x - 6x + 4 = 0
\]
Factor out \(x\):
\[
-(\sqrt{5} + 6)x + 4 = 0
\]
Solve for \(x\):
\[
(\sqrt{5} + 6)x = 4
\]
\[
x = \frac{4}{\sqrt{5} + 6}
\]

Thus, the solution is:
\[
x = \frac{-4}{\sqrt{5} - 6} \quad \text{or} \quad x = \frac{4}{\sqrt{5} + 6}
\]

Do you need further details or explanations on any of these steps?

Here are 5 questions to consider next:
1. How can you rationalize the denominators in both solutions?
2. What happens if \(\sqrt{5}\) is approximated numerically in the solutions?
3. Can the quadratic formula be applied to a different form of this equation?
4. What if the equation had a higher degree, such as a cubic term?
5. How do different absolute value cases affect the solutions?

**Tip:** When solving equations involving absolute values, always consider breaking the problem into cases based on the definition of \(|x|\).

Explore a detailed step-by-step solution to the equation √(5x²) - 4x + 4 = 2x. This problem involves algebraic concepts such as square root functions, absolute value, and piecewise equations. Suitable for high school students (Grades 10-12).

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