Question

The equation 4x²+8x-1=0 has its roots as alpha and beta.find the valuesof 1/alpha²+1/beta²?

Answer 1

`1/alpha^2+1/beta^2=72`

Explanation:

If the quadratic equation `ax^2+bx+c=0`, has roots `alpha and beta,`then `alpha+beta=-b/a and alpha*beta=c/a.`
So,
`4x^2+8x-1=0=>a=4,b=8,c=-1`
`=>alpha+beta=-8/4=-2 and alpha*beta=-1/4.`
Now,
`1/alpha^2+1/beta^2=(alpha^2+beta^2)/(alpha^2beta^2)=((alpha+beta)^2-2alphabeta)/(alphabeta)^2=((-2)^2-2(-1/4))/(-1/4)^2=(4+1/2)/(1/16)=(9/2)/(1/16)=9/cancel(2)*cancel(16)^8/1=72`