For what positive values of k , does the quadratic equation `3x^(2) - k x+ 3 = 0` not have real roots ?

Answer

To determine the positive values of \( k \) for which the quadratic equation \( 3x^2 - kx + 3 = 0 \) does not have real roots, we need to analyze the discriminant of the quadratic equation. The discriminant \( D \) is given by the formula:

\[ D = b^2 - 4ac \]

Where:
- \( a = 3 \) (coefficient of \( x^2 \))
- \( b = -k \) (coefficient of \( x \))
- \( c = 3 \) (constant term)

### Step 1: Write the expression for the discriminant
Substituting the values of \( a \), \( b \), and \( c \) into the discriminant formula, we get:

\[ D = (-k)^2 - 4 \cdot 3 \cdot 3 \]

### Step 2: Simplify the expression
Now, simplify the expression:

\[ D = k^2 - 36 \]

### Step 3: Set the condition for no real roots
For the quadratic equation to not have real roots, the discriminant must be less than zero:

\[ D < 0 \]

Substituting our expression for \( D \):

\[ k^2 - 36 < 0 \]

### Step 4: Solve the inequality
Rearranging the inequality gives:

\[ k^2 < 36 \]

Taking the square root of both sides, we find:

\[ -\sqrt{36} < k < \sqrt{36} \]

This simplifies to:

\[ -6 < k < 6 \]

### Step 5: Identify the positive values of \( k \)
Since we are looking for positive values of \( k \), we consider the range:

\[ 0 < k < 6 \]

### Final Answer
Thus, the positive values of \( k \) for which the quadratic equation \( 3x^2 - kx + 3 = 0 \) does not have real roots are:

\[ k \in (0, 6) \]

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