Question

The value of p for which the sum of the squares of the roots of the equation x² - (p - 2)x - p + 1 = 0 is

minimum will be

Answer 1

Correct Answer - Option 2 : 1

Formula used:

If α and β are the roots of quadratic equation ax2 + bx + c = 0

Sum of root α + β = -b/a

Product of root αβ = c/a

•  a2 + b2 = (a + b)2 - 2ab
• (a + b)² = a² + 2ab + b²
• (a - b)2 = a2 - 2ab + b2

 

Calculation:

Given that,

x2 - (p - 2)x - p + 1 = 0   ---(1)

Let the roots of equation are α and β. 

Sum of root α + β = (p - 2)

Product of root = αβ = -p + 1

∵ a² + b² = (a + b)² - 2ab

⇒  α² + β² = (p - 2)² - 2(-p + 1)

⇒ α2 + β2 = p2 - 4p + 4 + 2p - 2

⇒ α2 + β2 = (p - 1)² + 1  ---(2)

(p - 1)2 + 1  will be minimum when,

p = 1