1.

Find the values of k for which the roots are real and equal in each of the following equations:(i) 9x2 - 24x + k = 0(ii) 4x2 - 3kx + 1 = 0(iii) x2 - 2(5 + 2k)x+3(7 + 10k) = 0(iv) (3k + 1) x2 + 2 (k + 1) x + k = 0(v) kx2 + kx + 1 = -4x2 - x

Answer»

(i) 9x2 - 24x + k = 0

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

D = b2 – 4ac

If D = 0, roots are real and equal

9x2 - 24x + k = 0

⇒ D = 576 – 4 × 9 × k = 0

⇒ k = 576/36 = 16

(ii) 4x2 - 3kx + 1 = 0

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

D = b2 – 4ac

If D = 0, roots are real and equal

4x2 - 3kx + 1 = 0

⇒ D = 9k2 – 4 × 4 × 1 = 0

⇒ 9k2 = 16

⇒ k = 4/3

(iii) x2 - 2(5 + 2k)x + 3(7 + 10k) = 0

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

D = b2 – 4ac

If D = 0, roots are real and equal

x2 - 2(5 + 2k)x + 3(7 + 10k) = 0

⇒ 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

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

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

D = b2 – 4ac 

If D = 0, roots are real and equal

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

⇒ 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

(v) kx2 + kx + 1 = -4x2 - x

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

D = b2 – 4ac

If D = 0, roots are real and equal

kx2 + kx + 1 = -4x2 - x

⇒ (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



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