If `alpha` is a real root of the quadratic equation `a x^2+b x+c=0a n dbeta` ils a real root of ` a x^2+b x+c=0,` then show that there is a root `gamma` of equation `(a//2)x^2+b x+c=0` whilch lies between `aa n dbetadot`

Explanation

Let
`f(x) = (a)/(2) x^(2) + bx + c`
`rArr f(alpha) = (a)/(2) alpha^(2) + bx + c`
=` aalpha^(2) + balpha + c (a )/(2) alpha ^(2)`
` = -(a)/(2) alpha^(2) (becouse alpha` is root of `ax^(2) + bx + c = 0` )
and `f(beta) = (a)/(2) beta^(2) + b beta + c`
= - ` a beta^(2) + b beta + c + (3)/(2) a beta^(2)`
`= (3)/(2) a beta^(2) (becouse beta` is a root of `-ax^(2) + bx + c = 0 ) `
Now, `f(alpha)f(beta) = (-3)/(4) a^(2) alpha^(2) beta^(2) lt 0`
Hence, `f(x) = 0` has one real root between` alpha and beta` .