Question

Find the value of 'p' for which the equation px (x – 2) + 6 = 0 has two equal roots. Also, find the roots.

Answer 1

Given: Equation: px (x – 2) + 6 = 0 and we require two equal (double) roots.

Step-wise calculation:

1. Expand: px (x – 2) + 6 

= px² – 2px + 6

= 0

2. For a quadratic ax² + bx + c = 0, equal roots occur when discriminant D = b² – 4ac = 0.

Here a = p, b = –2p, c = 6.

3. Compute D: (–2p)² – 4(p)(6)

= 4p² – 24p

= 4p(p – 6)

4. Set D = 0

⇒ 4p(p – 6) = 0

⇒ p = 0 or p = 6

5. p = 0 makes the equation 6 = 0 (not a quadratic), so discard p = 0.

Thus p = 6 is the valid value.

6. Substitute p = 6: 6x² – 12x + 6 = 0

⇒ Divide by 6 

⇒ x² – 2x + 1 = 0

⇒ (x – 1)² = 0

⇒ Double root x = 1

p = 6 and the (equal) root is x = 1.