Solution:- Let F(x)=x^2+8x+5
We know that αβ = c/a
=5/1
=5
If α and β are roots of the equation x2+8x−5=0, then what is the quadratic equation whose roots are α/β and β/α?
Solution
Given equation is x^2+8x−5=0
α + β=-8
αβ=-5
We have to find the new Quadratic equation using this root α/β and β/α.
now sum of roots of new Quadratic Equation
α/β +β/α
=(α^2 + β^2)/αβ
=(α+β)^2/αβ
= (-8)^2/-5
=-64/5
product of roots
=(α/β)(β/α)
= 1
=5/5
here a=5,b=-64,c=5
we know that the standard form of Quadratic Equation
ax^2 +bx +c=0
5x^2 +(-64)x +5=0
5x^2 - 64x +5=0
the new Quadratic equation 5x^2 - 64x +5=0 .
If α and β are roots of equation x^2+5x+5=0 then write quadratic equation whose roots are α +1 and β+1.
Solution
Given equation is x^2+5x+5=0
α + β=-b/a=-5/1=-5
αβ=c/a=5/1=5
We have to find the new Quadratic equation using this root α/β and β/α.
now sum of roots of new Quadratic Equation
α +1 + β+1
=α+β+2
=-5+2
=-3
product of roots
=(α +1)(β+1)
=αβ+(α +β)+1
=5-5+1
=1
here a=1,b=-3,c=1
we know that the standard form of Quadratic Equation
ax^2 +bx +c=0
1x^2 +(-3)x+1=0
x^2 -3x +1=0
the new Quadratic equation x^2 -3x +1=0
Q.The roots α and β of the quadratic equation x^2 - 5x + 3(k - 1) = 0 are such that αβ = 1. Find the value k.
Solution
Given equation is x^2 - 5x + 3(k - 1) = 0
and αβ = 1
using formula,
c/a=1
3(k - 1)/1=1
k - 1=1/3
k =1/3+