Under what condition the sum of the roots of x^2 – mx + n = 0 is k times their
difference?​

1 thought on “Under what condition the sum of the roots of x^2 – mx + n = 0 is k times their<br />difference?​”

1. Step-by-step explanation:

   Given:–

   A quadratic equation is x^2 – mx + n = 0

   To find:–

   Under what condition the sum of the roots of

   x^2 – mx + n = 0 is k times their difference?

   Solution:–

   Given quadratic equation is x^2 – mx + n = 0

   On comparing with the standard quadratic equation ax^2 + bx + c = 0

   We have

   a = 1

   b= -m

   c = n

   Let α and β be the roots of the given equation

   (α > β)

   Sum of the roots = -b/a

   =>α + β = -(-m)/1

   =>α + β = m

   Sum of the roots = m————-(1)

   Product of the roots = c/a

   =>α × β = n/1

   =>αβ = n

   Diffrence of the roots = α – β

   We know that

   (a-b)^2=(a+b)^2-4ab

   =>(α – β)^2 = (α + β)^2 – 4αβ

   =>(α – β)^2 = m^2-4n

   =>α – β = √(m^2 -n)

   Difference between the roots = √(m^2 -n)—-(2)

   Sum of the roots = k× Difference of the roots

   From (1)&(2)

   =>m =K ×√(m^2 -n) (or)

   K = m/[√(m^2 -n)]

   Answer:–

   The required condition for the given problem is

   m =K ×√(m^2 -n) (or) K = m/[√(m^2 -n)]

   Used formulae:–

   • The standard quadratic equation is

   ax^2 + bx + c = 0

   • Sum of the roots = -b/a
   • Product of the roots = c/a
   • (a-b)^2=(a+b)^2-4ab

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