Question

If alpha is not equal to beta and difference between the roots of the polynomials x^2+ax+b and x^2+bx+a is the same then value of a+b=??

Answer 1

If a ≠ b and difference between the roots of the polynomials x² + ax + b and x² + bx + a is the same.

To find : The value of a + b

solution : let α and β are the roots of x² + bx + a.

sum of roots , (α + β) = -b/1 = -b

product of roots, αβ = a/1 = a

now difference of roots, |(α - β)| = √{(α + β)² - 4αβ}

= √{(-b)² - 4a} = √(b² - 4a)

similarly, let α' and β' are the roots of x² + ax + b.

sum of roots , (α' + β') = -a/1 = -a

product of roots, α'β' = b/1 = b

now difference of roots, |(α' - β')| = √{(α' + β')² - 4α'β'}

= √{(-a)² - 4b} = √(a² - 4b)

a/c to question,

|(α - β)| = |(α' - β')|

⇒√(b² - 4a) = √(a² - 4b)

⇒b² - 4a = a² - 4b

⇒b² - a² = 4a - 4b

⇒(b - a)(b + a) = 4(b - a)

⇒(b - a)[(b + a) - 4] = 0

as a ≠ b ⇒(b - a) ≠ 0

so, (b + a) - 4 = 0

⇒b + a = 4

Therefore the value of (a + b) = 4.