Find the values of k for which roots of the following equations are real and equal:
(i) `12x^(2)+4kx+3=0` (ii) `kx^(2)-5x+k=0`
(iii) `x^(2)+k(4x+k-1)+2=0` (iv) `x^(2)-2(5+2k)x+3(7+10k)=0`
(v) `5x^(2)-4x+2+k(4x^(2)-2x-1)=0` (vi) `(k+1)x^(2)-2(k-1)x+1=0`
(vii) `x^(2)-(3k-1)x+2k^(2)+2k-11=0` (viii) `2(k-12)x^(2)+2(k-12)x+2=0`

Answer

To find the values of \( k \) for which the roots of the given quadratic equations are real and equal, we need to set the discriminant \( D \) equal to zero. The discriminant \( D \) for a quadratic equation \( ax^2 + bx + c = 0 \) is given by:

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

If \( D = 0 \), the roots are real and equal. Let's solve each equation step by step.

### (i) \( 12x^2 + 4kx + 3 = 0 \)

1. Identify \( a = 12 \), \( b = 4k \), \( c = 3 \).
2. Calculate the discriminant:
\[ D = (4k)^2 - 4 \cdot 12 \cdot 3 \]
\[ D = 16k^2 - 144 \]
3. Set the discriminant equal to zero:
\[ 16k^2 - 144 = 0 \]
4. Solve for \( k^2 \):
\[ 16k^2 = 144 \]
\[ k^2 = \frac{144}{16} = 9 \]
5. Take the square root:
\[ k = \pm 3 \]

### (ii) \( kx^2 - 5x + k = 0 \)

1. Identify \( a = k \), \( b = -5 \), \( c = k \).
2. Calculate the discriminant:
\[ D = (-5)^2 - 4 \cdot k \cdot k \]
\[ D = 25 - 4k^2 \]
3. Set the discriminant equal to zero:
\[ 25 - 4k^2 = 0 \]
4. Solve for \( k^2 \):
\[ 4k^2 = 25 \]
\[ k^2 = \frac{25}{4} \]
5. Take the square root:
\[ k = \pm \frac{5}{2} \]

### (iii) \( x^2 + k(4x + k - 1) + 2 = 0 \)

1. Expand the equation:
\[ x^2 + (4k)x + (k^2 - k + 2) = 0 \]
2. Identify \( a = 1 \), \( b = 4k \), \( c = k^2 - k + 2 \).
3. Calculate the discriminant:
\[ D = (4k)^2 - 4 \cdot 1 \cdot (k^2 - k + 2) \]
\[ D = 16k^2 - 4(k^2 - k + 2) \]
\[ D = 16k^2 - 4k^2 + 4 - 8 \]
\[ D = 12k^2 - 4 \]
4. Set the discriminant equal to zero:
\[ 12k^2 - 4 = 0 \]
5. Solve for \( k^2 \):
\[ 12k^2 = 4 \]
\[ k^2 = \frac{4}{12} = \frac{1}{3} \]
6. Take the square root:
\[ k = \pm \frac{1}{\sqrt{3}} = \pm \frac{\sqrt{3}}{3} \]

### (iv) \( x^2 - 2(5 + 2k)x + 3(7 + 10k) = 0 \)

1. Identify \( a = 1 \), \( b = -2(5 + 2k) \), \( c = 3(7 + 10k) \).
2. Calculate the discriminant:
\[ D = [-2(5 + 2k)]^2 - 4 \cdot 1 \cdot 3(7 + 10k) \]
\[ D = 4(5 + 2k)^2 - 12(7 + 10k) \]
\[ D = 4(25 + 20k + 4k^2) - 84 - 120k \]
\[ D = 100 + 80k + 16k^2 - 84 - 120k \]
\[ D = 16k^2 - 40k + 16 \]
3. Set the discriminant equal to zero:
\[ 16k^2 - 40k + 16 = 0 \]
4. Solve using the quadratic formula:
\[ k = \frac{-(-40) \pm \sqrt{(-40)^2 - 4 \cdot 16 \cdot 16}}{2 \cdot 16} \]
\[ k = \frac{40 \pm \sqrt{1600 - 1024}}{32} \]
\[ k = \frac{40 \pm \sqrt{576}}{32} \]
\[ k = \frac{40 \pm 24}{32} \]
\[ k = \frac{64}{32} = 2 \quad \text{or} \quad k = \frac{16}{32} = \frac{1}{2} \]

### (v) \( 5x^2 - 4x + 2 + k(4x^2 - 2x - 1) = 0 \)

1. Combine like terms:
\[ (5 + 4k)x^2 + (-4 - 2k)x + (2 - k) = 0 \]
2. Identify \( a = 5 + 4k \), \( b = -4 - 2k \), \( c = 2 - k \).
3. Calculate the discriminant:
\[ D = (-4 - 2k)^2 - 4(5 + 4k)(2 - k) \]
\[ D = (16 + 16k + 4k^2) - 4(10 - 5k + 8k - 4k^2) \]
\[ D = 16 + 16k + 4k^2 - 40 + 20k - 32k + 16k^2 \]
\[ D = 20k^2 + 4k - 24 \]
4. Set the discriminant equal to zero:
\[ 20k^2 + 4k - 24 = 0 \]
5. Divide by 4:
\[ 5k^2 + k - 6 = 0 \]
6. Solve using the quadratic formula:
\[ k = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 5 \cdot (-6)}}{2 \cdot 5} \]
\[ k = \frac{-1 \pm \sqrt{1 + 120}}{10} \]
\[ k = \frac{-1 \pm \sqrt{121}}{10} \]
\[ k = \frac{-1 \pm 11}{10} \]
\[ k = 1 \quad \text{or} \quad k = -\frac{6}{5} \]

### (vi) \( (k+1)x^2 - 2(k-1)x + 1 = 0 \)

1. Identify \( a = k + 1 \), \( b = -2(k - 1) \), \( c = 1 \).
2. Calculate the discriminant:
\[ D = [-2(k - 1)]^2 - 4(k + 1)(1) \]
\[ D = 4(k^2 - 2k + 1) - 4(k + 1) \]
\[ D = 4k^2 - 8k + 4 - 4k - 4 \]
\[ D = 4k^2 - 12k \]
3. Set the discriminant equal to zero:
\[ 4k(k - 3) = 0 \]
4. Solve for \( k \):
\[ k = 0 \quad \text{or} \quad k = 3 \]

### (vii) \( x^2 - (3k - 1)x + (2k^2 + 2k - 11) = 0 \)

1. Identify \( a = 1 \), \( b = -(3k - 1) \), \( c = 2k^2 + 2k - 11 \).
2. Calculate the discriminant:
\[ D = (3k - 1)^2 - 4(1)(2k^2 + 2k - 11) \]
\[ D = (9k^2 - 6k + 1) - (8k^2 + 8k - 44) \]
\[ D = 9k^2 - 6k + 1 - 8k^2 - 8k + 44 \]
\[ D = k^2 - 14k + 45 \]
3. Set the discriminant equal to zero:
\[ k^2 - 14k + 45 = 0 \]
4. Solve using the quadratic formula:
\[ k = \frac{14 \pm \sqrt{(-14)^2 - 4 \cdot 1 \cdot 45}}{2 \cdot 1} \]
\[ k = \frac{14 \pm \sqrt{196 - 180}}{2} \]
\[ k = \frac{14 \pm \sqrt{16}}{2} \]
\[ k = \frac{14 \pm 4}{2} \]
\[ k = 9 \quad \text{or} \quad k = 5 \]

### (viii) \( 2(k - 12)x^2 + 2(k - 12)x + 2 = 0 \)

1. Factor out \( 2 \):
\[ 2[(k - 12)x^2 + (k - 12)x + 1] = 0 \]
2. Identify \( a = k - 12 \), \( b = k - 12 \), \( c = 1 \).
3. Calculate the discriminant:
\[ D = (k - 12)^2 - 4(k - 12)(1) \]
\[ D = (k - 12)^2 - 4(k - 12) \]
\[ D = (k - 12)((k - 12) - 4) = (k - 12)(k - 16) \]
4. Set the discriminant equal to zero:
\[ (k - 12)(k - 16) = 0 \]
5. Solve for \( k \):
\[ k = 12 \quad \text{or} \quad k = 16 \]

### Summary of Values of \( k \):
1. \( k = \pm 3 \)
2. \( k = \pm \frac{5}{2} \)
3. \( k = \pm \frac{\sqrt{3}}{3} \)
4. \( k = 2 \) or \( k = \frac{1}{2} \)
5. \( k = 1 \) or \( k = -\frac{6}{5} \)
6. \( k = 0 \) or \( k = 3 \)
7. \( k = 9 \) or \( k = 5 \)
8. \( k = 12 \) or \( k = 16 \)