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To solve the problem, we are given two equations:

1. \( u^2 - ut = 6 \)
2. \( t^2 - ut = -2 \)

We need to find the value of \( 2(u - t)^2 \).

First, let's isolate \(u^2\) and \(t^2\) from the given equations.

From the first equation:
\[ u^2 - ut = 6 \implies u^2 = ut + 6 \]

From the second equation:
\[ t^2 - ut = -2 \implies t^2 = ut - 2 \]

Now, we subtract the second equation from the first equation:
\[ u^2 - t^2 = (ut + 6) - (ut - 2) \]

Simplify:
\[ u^2 - t^2 = ut + 6 - ut + 2 \]
\[ u^2 - t^2 = 8 \]

We know that:
\[ u^2 - t^2 = (u + t)(u - t) \]

So:
\[ (u + t)(u - t) = 8 \]

Next, we need to find the value of \( 2(u - t)^2 \).

Let \( x = u - t \). Then \( (u + t)x = 8 \), so:
\[ u + t = \frac{8}{x} \]

The value of \( 2(u - t)^2 \) is \( 2x^2 \).

To find \( x \), we need another relationship. Consider:
\[ u^2 = ut + 6 \quad \text{and} \quad t^2 = ut - 2 \]

We already know \( u^2 - t^2 = 8 \).

Given the value \( u^2 - t^2 = (u + t)(u - t) \), substitute the value:
\[ (u + t)(u - t) = 8 \]

Let:
\[ a = u + t \]
\[ b = u - t \]

We know:
\[ ab = 8 \]

To find \( 2(u - t)^2 \):
\[ 2(u - t)^2 = 2b^2 \]

Next, solve for \( u \) and \( t \) using \( a \) and \( b \):
\[ u = \frac{a + b}{2} \]
\[ t = \frac{a - b}{2} \]

Given \( u^2 - ut = 6 \):
\[ \left(\frac{a + b}{2}\right)^2 - \left(\frac{a + b}{2}\right)\left(\frac{a - b}{2}\right) = 6 \]

Simplify:
\[ \frac{(a + b)^2}{4} - \frac{(a + b)(a - b)}{4} = 6 \]

Expand and simplify:
\[ \frac{a^2 + 2ab + b^2 - (a^2 - b^2)}{4} = 6 \]
\[ \frac{2ab + 2b^2}{4} = 6 \]
\[ \frac{2b(a + b)}{4} = 6 \]
\[ \frac{b(a + b)}{2} = 6 \]
\[ b(a + b) = 12 \]

We know \( ab = 8 \):
\[ a = \frac{8}{b} \]

Substitute into the equation:
\[ b\left(\frac{8}{b} + b\right) = 12 \]
\[ 8 + b^2 = 12 \]
\[ b^2 = 4 \]
\[ b = 2 \text{ or } b = -2 \]

Thus:
\[ 2(u - t)^2 = 2b^2 = 2(2^2) = 2 \cdot 4 = 8 \]

Therefore, the value is:
\[ \boxed{8} \]

Would you like details or have any questions?

Here are 8 related questions:
1. How would you solve for \(u\) and \(t\) individually?
2. Can you verify the solution using another method?
3. How do the equations change if one of the constants is altered?
4. What happens if both \(u\) and \(t\) are negative?
5. How does this problem relate to solving quadratic equations?
6. What are the geometric interpretations of these equations?
7. How can these techniques be applied to other algebraic problems?
8. What is the effect of squaring both sides of an equation?

**Tip:** Always check your solutions by substituting back into the original equations.

Learn how to solve the quadratic equations u^2 - ut = 6 and t^2 - ut = -2 with detailed step-by-step solutions. This problem involves algebraic manipulation and application of the quadratic formula. Suitable for high school students and those preparing for algebra exams.

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