The value of k for which the equation `x^(2)-(3k-1)x+2k^(2)+2k=11` have equal roots, is

Options

A. 5
B. 9
C. both (a) and (b)
D. 0

Answer

To find the value of \( k \) for which the equation

\[ x^2 - (3k - 1)x + (2k^2 + 2k) = 11 \]

has equal roots, we need to set the discriminant of the quadratic equation to zero. The discriminant \( D \) is given by the formula:

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

### Step 1: Rearranging the equation
First, we rearrange the equation to standard quadratic form:

\[ x^2 - (3k - 1)x + (2k^2 + 2k - 11) = 0 \]

Here, we identify:
- \( a = 1 \)
- \( b = -(3k - 1) = 1 - 3k \)
- \( c = 2k^2 + 2k - 11 \)

### Step 2: Setting the discriminant to zero
For the roots to be equal, we set the discriminant \( D \) to zero:

\[ D = (1 - 3k)^2 - 4(1)(2k^2 + 2k - 11) = 0 \]

### Step 3: Expanding the discriminant
Now we expand \( D \):

\[ (1 - 3k)^2 = 1 - 6k + 9k^2 \]

And for the second part:

\[ -4(2k^2 + 2k - 11) = -8k^2 - 8k + 44 \]

### Step 4: Combining the terms
Now we combine these results:

\[ 1 - 6k + 9k^2 - 8k^2 - 8k + 44 = 0 \]

This simplifies to:

\[ (9k^2 - 8k^2) + (-6k - 8k) + (1 + 44) = 0 \]

\[ k^2 - 14k + 45 = 0 \]

### Step 5: Factoring the quadratic
Next, we factor the quadratic equation:

\[ k^2 - 14k + 45 = 0 \]

We look for two numbers that multiply to \( 45 \) and add to \( -14 \). The numbers are \( -9 \) and \( -5 \):

\[ (k - 9)(k - 5) = 0 \]

### Step 6: Finding the values of \( k \)
Setting each factor to zero gives us:

\[ k - 9 = 0 \quad \Rightarrow \quad k = 9 \]
\[ k - 5 = 0 \quad \Rightarrow \quad k = 5 \]

### Conclusion
Thus, the values of \( k \) for which the equation has equal roots are \( k = 5 \) and \( k = 9 \).