1.

Find the values of k for which the roots are real and equal in each of the following equations:(i) (k + 1) x2 + 2 (k + 3) x + (k + 8) = 0(ii) x2 - 2kx + 7x + 1/4 = 0(iii) (k + 1) x2 - 2 (3k + 1) x + 8k + 1 = 0(iv) 5x2 - 4x + 2 +k ((4x2 - 2x - 1) = 0(v) (4 - k) x2 + (2k + 4) x + (8k + 1) = 0

Answer»

(i) (k + 1) x2 + 2 (k + 3) x + (k + 8) = 0

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

D = b2 – 4ac

If D = 0, roots are real and equal

(k + 1) x2 + 2 (k + 3) x + (k + 8) = 0

⇒ D = 4(k + 3)2 – 4(k + 1)(k + 8) = 0 

⇒ 4k2 + 36 + 24k – 4k2 – 32 – 36k = 0 

⇒ 12k = 4 

⇒ k = 1/3

(ii) 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

(iii) (k + 1)x2 - 2(3k + 1)x + 8k + 1 = 0

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

D = b2 – 4ac

If D = 0, roots are real and equal

(k + 1)x2 - 2(3k + 1)x + 8k + 1 = 0

⇒ 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

(iv) 5x2 - 4x + 2 + k (4x2 - 2x - 1) = 0

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

D = b2 – 4ac

If D = 0, roots are real and equal

5x2 - 4x + 2 + k (4x2 - 2x - 1) = 0

⇒ (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

(v) (4 - k)x2 + (2k + 4)x + (8k + 1) = 0

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

D = b2 – 4ac

If D = 0, roots are real and equal

(4 - k)x2 + (2k + 4)x + (8k + 1) = 0

⇒ 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



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