If the cubic `2x^3-9x^2+12 x+k=0` has two equal roots then minimum value of `|k|` is______.

Explanation

To solve the problem, we need to find the minimum value of \(|k|\) for the cubic equation \(2x^3 - 9x^2 + 12x + k = 0\) when it has two equal roots. Let's go through the solution step by step.

### Step 1: Understanding the Condition for Equal Roots
For a cubic polynomial to have two equal roots, the discriminant must be zero. We can express the roots as \(\alpha, \alpha, \beta\), where \(\alpha\) is the repeated root.

### Step 2: Using Vieta's Formulas
From Vieta's formulas, we know:
1. The sum of the roots:
\[ 2\alpha + \beta = \frac{9}{2} \]
2. The sum of the products of the roots taken two at a time:
\[ \alpha^2 + 2\alpha\beta = 6 \]
3. The product of the roots:
\[ \alpha^2\beta = -\frac{k}{2} \]

### Step 3: Express \(\beta\) in Terms of \(\alpha\)
From the first equation:
\[ \beta = \frac{9}{2} - 2\alpha \]

### Step 4: Substitute \(\beta\) into the Second Equation
Substituting \(\beta\) into the second equation:
\[ \alpha^2 + 2\alpha\left(\frac{9}{2} - 2\alpha\right) = 6 \]
Expanding this gives:
\[ \alpha^2 + 9\alpha - 4\alpha^2 = 6 \]
Combining like terms:
\[ -3\alpha^2 + 9\alpha - 6 = 0 \]
Multiplying through by -1:
\[ 3\alpha^2 - 9\alpha + 6 = 0 \]

### Step 5: Solve the Quadratic Equation
Using the quadratic formula \(\alpha = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[ \alpha = \frac{9 \pm \sqrt{(-9)^2 - 4 \cdot 3 \cdot 6}}{2 \cdot 3} \]
Calculating the discriminant:
\[ \sqrt{81 - 72} = \sqrt{9} = 3 \]
Thus,
\[ \alpha = \frac{9 \pm 3}{6} \]
This gives:
\[ \alpha = 2 \quad \text{or} \quad \alpha = 1 \]

### Step 6: Find Corresponding \(\beta\) Values
1. If \(\alpha = 2\):
\[ \beta = \frac{9}{2} - 2(2) = \frac{9}{2} - 4 = \frac{1}{2} \]
2. If \(\alpha = 1\):
\[ \beta = \frac{9}{2} - 2(1) = \frac{9}{2} - 2 = \frac{5}{2} \]

### Step 7: Calculate \(k\) for Each Case
Using the product of the roots:
1. For \(\alpha = 2\) and \(\beta = \frac{1}{2}\):
\[ \alpha^2\beta = 2^2 \cdot \frac{1}{2} = 2 \quad \Rightarrow \quad -\frac{k}{2} = 2 \quad \Rightarrow \quad k = -4 \]
2. For \(\alpha = 1\) and \(\beta = \frac{5}{2}\):
\[ \alpha^2\beta = 1^2 \cdot \frac{5}{2} = \frac{5}{2} \quad \Rightarrow \quad -\frac{k}{2} = \frac{5}{2} \quad \Rightarrow \quad k = -5 \]

### Step 8: Find Minimum Value of \(|k|\)
The values of \(k\) we found are \(-4\) and \(-5\). Therefore, the minimum value of \(|k|\) is:
\[ \min(|-4|, |-5|) = \min(4, 5) = 4 \]

### Final Answer
The minimum value of \(|k|\) is \(4\).

---