The value of k for which equations `x^(2)+(2k-6)x+7-3k=0 and x^(2)+(2k-2)x+3k-5=0` has two different pair of equal roots is

Answer

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

1. \( x^2 + (2k - 6)x + (7 - 3k) = 0 \)
2. \( x^2 + (2k - 2)x + (3k - 5) = 0 \)

have two different pairs of equal roots, we need to ensure that both equations have equal roots. This occurs when the discriminant of each equation is equal to zero.

### Step 1: Calculate the discriminant of the first equation

The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:

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

For the first equation, we have:

- \( a = 1 \)
- \( b = 2k - 6 \)
- \( c = 7 - 3k \)

Thus, the discriminant \( D_1 \) is:

\[ D_1 = (2k - 6)^2 - 4 \cdot 1 \cdot (7 - 3k) \]

Expanding this:

\[ D_1 = (2k - 6)^2 - 4(7 - 3k) \]
\[ = 4k^2 - 24k + 36 - (28 - 12k) \]
\[ = 4k^2 - 24k + 36 - 28 + 12k \]
\[ = 4k^2 - 12k + 8 \]

Setting the discriminant equal to zero for equal roots:

\[ 4k^2 - 12k + 8 = 0 \]

### Step 2: Simplify the equation

Dividing the entire equation by 4:

\[ k^2 - 3k + 2 = 0 \]

### Step 3: Factor the quadratic equation

Factoring gives:

\[ (k - 1)(k - 2) = 0 \]

Thus, the possible values for \( k \) are:

\[ k = 1 \quad \text{or} \quad k = 2 \]

### Step 4: Calculate the discriminant of the second equation

Now, we will do the same for the second equation:

For the second equation, we have:

- \( a = 1 \)
- \( b = 2k - 2 \)
- \( c = 3k - 5 \)

Thus, the discriminant \( D_2 \) is:

\[ D_2 = (2k - 2)^2 - 4 \cdot 1 \cdot (3k - 5) \]

Expanding this:

\[ D_2 = (2k - 2)^2 - 4(3k - 5) \]
\[ = 4k^2 - 8k + 4 - (12k - 20) \]
\[ = 4k^2 - 8k + 4 - 12k + 20 \]
\[ = 4k^2 - 20k + 24 \]

Setting the discriminant equal to zero for equal roots:

\[ 4k^2 - 20k + 24 = 0 \]

### Step 5: Simplify the equation

Dividing the entire equation by 4:

\[ k^2 - 5k + 6 = 0 \]

### Step 6: Factor the quadratic equation

Factoring gives:

\[ (k - 2)(k - 3) = 0 \]

Thus, the possible values for \( k \) are:

\[ k = 2 \quad \text{or} \quad k = 3 \]

### Step 7: Find the common value of \( k \)

Now, we have two sets of values for \( k \):

1. From the first equation: \( k = 1 \) or \( k = 2 \)
2. From the second equation: \( k = 2 \) or \( k = 3 \)

The common value that satisfies both equations is:

\[ \boxed{2} \]