Product of roots of x^2+8x+5

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+1

k=(1+3)/3

k=4/3 Ans

 

The roots α  and β  of the quadratic equation x2−5x+3(k−1)=0  are such that α−β=11 Find the value of k.