Question

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 1

(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