Find the values of k for which the roots are real and equal in each of the following equations:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
(vii) 
(viii) 
(ix) 
(x) 
(xi) 
(xii) x2 – 2kx + 7x + 1/4 = 0
(xiii) 
(xiv) 
(xv) 
(xvi) 
(xvii) 
(xviii) 
(xix) 
(xx) 
(xxi) 
(xxii) 
(xxiii) 
(xxiv) 
(xxv) 
(xxvi) 
(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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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