Answer :

(i)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D = 0, roots are real and equal



D = 16 – 4k = 0


k = 4


(ii)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D = 0, roots are real and equal



D = 4 × 5 – 4 × 4k = 0


k = 5/4


(iii)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D = 0, roots are real and equal



D = 25 – 4 × 3 × 2k = 0


k = 25/24


(iv)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D = 0, roots are real and equal



D = k2 – 4 × 4 × 9 = 0


k2 – 144 = 0


k = �12


(v)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D = 0, roots are real and equal



1600 – 4 × 2k × 25 = 0


k = 8


(vi)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D = 0, roots are real and equal



D = 576 – 4 × 9 × k = 0


k = 576/36 = 16


(vii)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D = 0, roots are real and equal



D = 9k2 – 4 × 4 × 1 = 0


9k2 = 16


k = �4/3


(viii)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D = 0, roots are real and equal



D = 4(5 + 2k)2 – 4 × 3(7 + 10k) = 0


100 + 16k2 + 80k – 84 – 120k = 0


16k2 – 40k + 16 = 0


2k2 – 5k + 2 = 0


2k2 – 4k – k + 2 = 0


2k(k – 2) – (k – 2) = 0


(2k – 1)(k – 2) = 0


k = 2, 1/2


(ix)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D = 0, roots are real and equal



D = 4(k + 1)2 – 4k(3k + 1) = 0


4k2 + 8k + 4 – 12k2 – 4k = 0


2k2 – k – 1 = 0


2k2 – 2k + k – 1 = 0


2k(k – 1) + (k – 1) = 0


(2k + 1)(k – 1) = 0


k = 1, -1/2


(x)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D = 0, roots are real and equal



(k + 4)x2 + (k + 1)x + 1 = 0


D = (k + 1)2 – 4(k + 4) = 0


k2 + 2k + 1 – 4k – 16 = 0


k2 – 2k – 15 = 0


k2 – 5k + 3k – 15 = 0


k(k – 5) + 3(k – 5) = 0


(k + 3)(k – 5) = 0


k = 5, -3


(xi)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D = 0, roots are real and equal



D = 4(k + 3)2 – 4(k + 1)(k + 8) = 0


4k2 + 36 + 24k – 4k2 – 32 – 36k = 0


12k = 4


k = 1/3


(xii) x2 – 2kx + 7x + 1/4 = 0


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D = 0, roots are real and equal


x2 – 2kx + 7x + 1/4 = 0


D = (7 – 2k)2 – 4 × 1/4 = 0


49 + 4k2 – 28k – 1 = 0


k2 – 7k + 12 = 0


k2 – 4k – 3k + 12 = 0


k(k – 4) – 3(k – 4) = 0


(k – 3)(k – 4) = 0


k = 3, 4


(xiii)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D = 0, roots are real and equal



D = 4(3k + 1)2 – 4(k + 1)(8k + 1) = 0


4 × (9k2 + 6k + 1) – 32k2 – 4 – 36k = 0


36k2 + 24k + 4 – 32k2 – 4 – 36k = 0


4k(k – 3) = 0


k = 0, 3


(xiv)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D = 0, roots are real and equal



(5 + 4k)x2 – (4 + 2k)x + 2 – k = 0


D = (4 + 2k)2 – 4 × (5 + 4k)(2 – k) = 0


16 + 4k2 + 16k + 16k2 – 12k – 40 = 0


20k2 – 4k – 24 = 0


5k2 - k - 6 = 0


5k2 – 6k + 5k – 6 = 0


k(5k – 6) + (5k – 6) = 0


(k + 1)(5k – 6) = 0


k = -1, 6/5


(xv)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D = 0, roots are real and equal



D = (2k + 4)2 – 4 × (4 – k)(8k + 1) = 0


4k2 + 16 + 16k + 32k2 – 16 – 124k = 0


36k2 – 108k = 0


36k(k – 3) = 0


k = 0, 3


(xvi)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D = 0, roots are real and equal



D = 4(k + 3)2 – 4 × (2k + 1)(k + 5) = 0


4k2 + 36 + 24k – 8k2 – 20 – 44k = 0


-4k2 – 20k + 16 = 0


k2 + 5k – 4 = 0



(xvii)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D = 0, roots are real and equal



D = 4(k + 1)2 – 4 × 4(k + 4) = 0


4k2 + 8k + 4 – 16k – 64 = 0


k2 – 2k - 15 = 0


k2 – 5k + 3k – 15 = 0


(k – 5)(k + 3) = 0


k = -3, 5


(xviii)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D = 0, roots are real and equal



D = 4(k + 1)2 – 4k2 = 0


4k2 + 8k + 4 – 4k2 = 0


k = -1/2


(xix)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D = 0, roots are real and equal



D = 4(k – 1)2 – 4 × 4k2 = 0


4k2 – 8k + 4 – 16k2 = 0


12k2 + 8k – 4 = 0


3k2 + 2k – 1 = 0


3k2 + 3k – k – 1 = 0


3k(k + 1) –(k + 1) = 0


(3k – 1)(k + 1) = 0


k = 1/3, - 1


(xx)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D = 0, roots are real and equal



D = 4(k – 1)2 – 4 × (k + 1) = 0


4k2 – 8k + 4 – 4k – 4 = 0


4k(k – 3) = 0


k = 0, 3


(xxi)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D = 0, roots are real and equal



D = k2 – 4 × 2 × 3 = 0


k2 = = 24


k = �2√6


(xxii)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D = 0, roots are real and equal



kx2 – 2kx + 6 = 0


D = 4k2 – 4 × 6 × k = 0


4k(k – 6) = 0


k = 0, 6 but k can’t be 0 a it is the coefficient of x2, thus k = 6


(xxiii)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D = 0, roots are real and equal



D = 16k2 – 4k = 0


4k(4k – 1) = 0


k = 0, 1/4


(xxiv)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D = 0, roots are real and equal



kx2 – 2√5kx + 10 = 0


D = 4 × 5k2 – 4 × k × 10 = 0


k2 = 2k


k = 2


(xxv)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D = 0, roots are real and equal



px2 – 3px + 9 = 0


D = 9p2 – 4 × 9 × p = 0


p = 4


(xxvi)


For a quadratic equation, ax2 + bx + c = 0,


D = b2 – 4ac


If D = 0, roots are real and equal



D = p2 – 4 × 4 × 3 = 0


p2 = 48


p = �4√3


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