2x$$ \left(4-x\right)^{-\frac12}-3\left(\sqrt{4-x}\right)=0 $$

We are given the equation:

\[
2x \left(4 - x\right)^{-\frac{1}{2}} - 3\left(\sqrt{4 - x}\right) = 0
\]

### Step 1: Substitute a new variable
Let's simplify this by substituting $ y = \sqrt{4 - x} $, so $ y^2 = 4 - x $ and $ x = 4 - y^2 $.

Substitute these into the equation:

\[
2(4 - y^2) y^{-1} - 3y = 0
\]

### Step 2: Simplify the equation
Now, simplify the terms:

\[
2(4 - y^2) \cdot \frac{1}{y} - 3y = 0
\]

\[
\frac{2(4 - y^2)}{y} - 3y = 0
\]

Expand the first term:

\[
\frac{8 - 2y^2}{y} - 3y = 0
\]

### Step 3: Eliminate the fraction
Multiply the entire equation by $ y $ to eliminate the denominator:

\[
8 - 2y^2 - 3y^2 = 0
\]

### Step 4: Combine like terms
Simplify the quadratic expression:

\[
8 - 5y^2 = 0
\]

### Step 5: Solve for $ y^2 $
Rearrange the equation:

\[
5y^2 = 8
\]

\[
y^2 = \frac{8}{5}
\]

### Step 6: Solve for $ y $
Take the square root of both sides:

\[
y = \pm \sqrt{\frac{8}{5}} = \pm \frac{2\sqrt{10}}{5}
\]

### Step 7: Solve for $ x $
Recall that $ y = \sqrt{4 - x} $. Substituting $ y = \pm \frac{2\sqrt{10}}{5} $:

\[
\sqrt{4 - x} = \pm \frac{2\sqrt{10}}{5}
\]

Square both sides:

\[
4 - x = \frac{8}{5}
\]

Solve for $ x $:

\[
x = 4 - \frac{8}{5} = \frac{20}{5} - \frac{8}{5} = \frac{12}{5}
\]

Thus, the solution is:

\[
x = \frac{12}{5}
\]

Let me know if you want more details or further explanation.

Here are 5 related questions:
1. How does substitution help simplify complex equations?
2. What is the importance of eliminating fractions in an equation?
3. Why do we square both sides when solving equations involving square roots?
4. What does it mean for a solution to have both positive and negative square roots?
5. How can we verify the solution by substituting it back into the original equation?

**Tip**: When solving equations involving square roots, always check if both the positive and negative roots provide valid solutions in the context of the problem.

2x \left(4 - x\right)^{-\frac{1}{2}} - 3\left(\sqrt{4 - x}\right) = 0

Learn how to solve the equation 2x(4-x)^(-1/2) - 3√(4-x) = 0 using substitution. This problem involves algebraic concepts such as quadratic equations and square roots. Detailed steps are provided for high school-level math students in Grades 10-12.

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