If `y ^(2)(y^(2) -6) + x ^(2) -8x +24 =0 ` and the minimum value of `x ^(2) + y^(4)` is m and maximum value is M, then find the value of `M-2m.`

Answer

To solve the problem, we start with the given equation:

\[ y^2(y^2 - 6) + x^2 - 8x + 24 = 0 \]

### Step 1: Rearranging the Equation
First, we can rewrite the equation in a more manageable form:

\[ y^4 - 6y^2 + x^2 - 8x + 24 = 0 \]

### Step 2: Substituting Variables
Let \( t = y^2 \). Then the equation becomes:

\[ t^2 - 6t + x^2 - 8x + 24 = 0 \]

### Step 3: Rearranging Further
Rearranging the equation gives us:

\[ t^2 + x^2 - 6t - 8x + 24 = 0 \]

### Step 4: Completing the Square
Next, we complete the square for the \( x \) and \( t \) terms:

- For \( x^2 - 8x \):
\[ x^2 - 8x = (x - 4)^2 - 16 \]

- For \( t^2 - 6t \):
\[ t^2 - 6t = (t - 3)^2 - 9 \]

Substituting these back into the equation gives:

\[ (t - 3)^2 - 9 + (x - 4)^2 - 16 + 24 = 0 \]

Simplifying this, we have:

\[ (t - 3)^2 + (x - 4)^2 - 1 = 0 \]

### Step 5: Circle Equation
This can be rewritten as:

\[ (t - 3)^2 + (x - 4)^2 = 1 \]

This represents a circle centered at \( (4, 3) \) with a radius of \( 1 \).

### Step 6: Finding Minimum and Maximum Values
We need to find the minimum and maximum values of \( x^2 + y^4 \). Recall that \( y^4 = t^2 \), so we need to minimize and maximize:

\[ x^2 + t^2 \]

### Step 7: Distance from Origin
The distance from the origin \( O(0, 0) \) to the center of the circle \( C(4, 3) \) is:

\[ OC = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = 5 \]

### Step 8: Minimum Value Calculation
The minimum distance from the origin to the circle is:

\[ OC - \text{radius} = 5 - 1 = 4 \]

Thus, the minimum value of \( x^2 + t^2 \) is:

\[ (4)^2 = 16 \]

### Step 9: Maximum Value Calculation
The maximum distance from the origin to the circle is:

\[ OC + \text{radius} = 5 + 1 = 6 \]

Thus, the maximum value of \( x^2 + t^2 \) is:

\[ (6)^2 = 36 \]

### Step 10: Final Calculation
Let \( m \) be the minimum value and \( M \) be the maximum value:

\[ m = 16, \quad M = 36 \]

We need to find \( M - 2m \):

\[ M - 2m = 36 - 2 \times 16 = 36 - 32 = 4 \]

### Final Answer
Thus, the value of \( M - 2m \) is:

\[ \boxed{4} \]