Question

If –4 is a root of the equation x² + 2x + 4p = 0, find the value of k for which the quadratic equation x² + px(1 + 3k) + 7(3 + 2k) = 0 has equal roots.

Answer 1

It is given that –4 is a root of the quadratic equation x² + 2x + 4p = 0

∴ (–4)² + 2 × (–4) + 4p = 0 

⇒ 16 – 8 + 4p = 0 

⇒ 4p + 8 = 0 

⇒ p = –2 

The equation x² + px(1 + 3k) + 7(3 + 2k) = 0 has real roots 

∴ D = 0 

⇒ [p(1 + 3k)]² – 4 × 1 × 7(3 + 2k) = 0 

⇒ [–2(1 + 3k)]² – 28(3 + 2k) = 0  

⇒ 4(1 + 6k + 9k²) – 28(3 + 2k) = 0 

⇒ 4(1 + 6k + 9k² – 21 – 14k) = 0 

⇒ 9k² – 8k – 20 = 0 

⇒ 9k² – 18k + 10k – 20 = 0 

⇒ 9k(k – 2) + 10(k – 2) = 0 

⇒ (k – 2)(9k + 10) = 0 

⇒ k – 2 = 0 or 9k + 10 = 0 

⇒ k = 2 or k = `(-10)/9` 

Hence, the required value of k is 2 or `(-10)/9`.