If y²(y²-6) + x²-8x+24=0 and the minimum value of x²+y⁴ is m and the maximum value is M; then find the value of M-2m

To solve the problem, we need to analyze the given equation and find the minimum and maximum values of the expression \(x^2 + y^4\). The equation provided is:

y²(y² - 6) + x² - 8x + 24 = 0.

Let's start by simplifying this equation. We can rewrite the quadratic part in terms of \(x\) and \(y\). The equation can be rearranged as:

y^4 - 6y² + x² - 8x + 24 = 0.

Next, we can complete the square for the \(x\) terms:

x² - 8x = (x - 4)² - 16.

Substituting this back into the equation gives us:

y^4 - 6y² + (x - 4)² - 16 + 24 = 0,

which simplifies to:

y^4 - 6y² + (x - 4)² + 8 = 0.

Now, we can rearrange this to isolate the squared term:

(x - 4)² = -y^4 + 6y² - 8.

For the right side to be non-negative (since a square cannot be negative), we need:

-y^4 + 6y² - 8 ≥ 0.

This can be rewritten as:

y^4 - 6y² + 8 ≤ 0.

Let’s denote \(z = y²\). The inequality becomes:

z² - 6z + 8 ≤ 0.

To find the roots of the quadratic equation \(z² - 6z + 8 = 0\), we can use the quadratic formula:

z = \frac{-b ± \sqrt{b² - 4ac}}{2a} = \frac{6 ± \sqrt{(-6)² - 4(1)(8)}}{2(1)} = \frac{6 ± \sqrt{36 - 32}}{2} = \frac{6 ± 2}{2}.

This gives us the roots:

• z = 4
• z = 2

Thus, the inequality \(z² - 6z + 8 ≤ 0\) holds for \(2 ≤ z ≤ 4\). Since \(z = y²\), we have:

2 ≤ y² ≤ 4.

This implies:

• √2 ≤ |y| ≤ 2.

Now, we can find the corresponding values of \(x\) using the equation:

(x - 4)² = -y^4 + 6y² - 8.

We will evaluate this expression at the endpoints of the interval for \(y²\). First, let’s calculate for \(y² = 2\):

y² = 2 gives us:

(x - 4)² = -2² + 6(2) - 8 = -4 + 12 - 8 = 0.

Thus, \(x - 4 = 0\) or \(x = 4\).

Next, for \(y² = 4\):

y² = 4 gives us:

(x - 4)² = -4² + 6(4) - 8 = -16 + 24 - 8 = 0.

Again, \(x - 4 = 0\) or \(x = 4\).

Now, we can find the values of \(x² + y^4\) at these points:

For \(y² = 2\) (which means \(y = ±\sqrt{2}\)):

x² + y^4 = 4² + 2² = 16 + 4 = 20.

For \(y² = 4\) (which means \(y = ±2\)):

x² + y^4 = 4² + 4² = 16 + 16 = 32.

Thus, the minimum value \(m = 20\) and the maximum value \(M = 32\). Now, we can find \(M - 2m\):

M - 2m = 32 - 2(20) = 32 - 40 = -8.

Therefore, the final answer is:

-8.