Question

If 3 is a root of the quadratic equation x² – x + k = 0, find the value of p so that the roots of the equation x² + 2kx + (k² + 2k + p) = 0 are equal.

If 3 is a root of the quadratic equation x² – x + k = 0, find the value of p so that the roots of the equation x² + k(2x + k + 2) + p = 0 are equal.

Answer 1

It is given that 3 is a root of the quadratic equation x² – x + k = 0 

∴ (3)² – 3 + k = 0 

⇒ k + 6 = 0 

⇒ k = –6 

The roots of the equation x² + 2kx + (k² + 2k + p) = 0 are equal. 

∴ D = 0 

⇒ (2k)² – 4 × 1 × (k² + 2k + p) = 0 

⇒ 4k² – 4k² – 8k – 4p = 0 

⇒ –8k – 4p = 0 

⇒ `p = (8k)/-4 = -2k` 

⇒ p = –2 × (–6) = 12 

Hence, the value of p is 12.