If `lambda` be the ratio of the roots of the quadratic equation in x, `3m^2 x^2+m(m-4)x+2=0`, then the least value of m for which `lambda+(1)/(lambda)=1`, is `k-3 sqrt(2)`. The value of k is __________.

Explanation

To solve the problem, we need to find the least value of \( m \) for which the ratio of the roots \( \lambda + \frac{1}{\lambda} = 1 \) holds true for the quadratic equation given. Let's break down the solution step by step.

### Step 1: Identify the coefficients of the quadratic equation
The quadratic equation is given as:
\[ 3m^2 x^2 + m(m-4)x + 2 = 0 \]
Here, we can identify:
- \( a = 3m^2 \)
- \( b = m(m-4) \)
- \( c = 2 \)

### Step 2: Use the relationships for the roots
Let \( \alpha \) and \( \beta \) be the roots of the quadratic equation. According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} = -\frac{m(m-4)}{3m^2} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} = \frac{2}{3m^2} \)

### Step 3: Express \( \lambda \)
The ratio of the roots \( \lambda \) is defined as:
\[ \lambda = \frac{\alpha}{\beta} \]
Using the relationships from Vieta's formulas, we can express \( \lambda \) in terms of \( \alpha + \beta \) and \( \alpha \beta \).

### Step 4: Set up the equation
From the condition given in the problem:
\[ \lambda + \frac{1}{\lambda} = 1 \]
Substituting \( \lambda \) gives:
\[ \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = 1 \]
Multiplying through by \( \alpha \beta \):
\[ \alpha^2 + \beta^2 = \alpha \beta \]
Using the identity \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \):
\[ (\alpha + \beta)^2 - 2\alpha\beta = \alpha \beta \]
This simplifies to:
\[ (\alpha + \beta)^2 = 3\alpha \beta \]

### Step 5: Substitute the values
Substituting the expressions for \( \alpha + \beta \) and \( \alpha \beta \):
\[ \left(-\frac{m(m-4)}{3m^2}\right)^2 = 3 \cdot \frac{2}{3m^2} \]
Squaring the left side gives:
\[ \frac{m^2(m-4)^2}{9m^4} = \frac{2}{m^2} \]

### Step 6: Cross-multiply and simplify
Cross-multiplying leads to:
\[ m^2(m-4)^2 = 18 \]
Expanding the left side:
\[ m^2(m^2 - 8m + 16) = 18 \]
This simplifies to:
\[ m^4 - 8m^3 + 16m^2 - 18 = 0 \]

### Step 7: Solve the polynomial
To find the least value of \( m \), we can use the quadratic formula or factorization. However, we can also analyze the roots directly:
\[ m^4 - 8m^3 + 16m^2 - 18 = 0 \]
Using the quadratic formula for \( m^2 \):
Let \( y = m^2 \):
\[ y^2 - 8y + 16 - 18 = 0 \implies y^2 - 8y - 2 = 0 \]
Using the quadratic formula:
\[ y = \frac{8 \pm \sqrt{64 + 8}}{2} = \frac{8 \pm \sqrt{72}}{2} = 4 \pm 3\sqrt{2} \]
Thus, \( m^2 = 4 + 3\sqrt{2} \) or \( m^2 = 4 - 3\sqrt{2} \).

### Step 8: Find \( m \)
Taking the square root:
- \( m = \sqrt{4 + 3\sqrt{2}} \) (positive)
- \( m = \sqrt{4 - 3\sqrt{2}} \) (check if positive)

### Step 9: Determine the least value of \( m \)
The least value occurs at:
\[ m = 4 - 3\sqrt{2} \]
Given that \( k - 3\sqrt{2} = 4 - 3\sqrt{2} \), we find \( k = 4 \).

### Final Answer
Thus, the value of \( k \) is:
\[ \boxed{4} \]