Under what condition the sum of the roots of x^2 – mx + n = 0 is k times theirdifference? About the author Mary
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 Reply
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:–
ax^2 + bx + c = 0