Question

If ∝ and β are the roots of the equation a$x^2$ + bx + c = 0, (a,b,c ​ R) , then (1+α+${\alpha }^2$) (1+β+${\beta }^2$) is :

• A. < 0
• B. >0
• C. =0
• D. None of these

Answer 1

The correct option is B

>0

α+β= -b/a , αβ= c/a
(1+α+${\alpha }^2$) (1+β+${\beta }^2$)
= 1+(α+β)+(${\alpha }^2$+${\beta }^2$)+αβ+αβ(α+β)+${\alpha }^2{\beta }^2$
=1+(α+β)+${(\alpha +\beta )}^2$ – αβ+ αβ(α+β)+${\left(\alpha \beta \right)}^2$
=1-b/a + $\frac{b^2}{a^2}$- $\frac{c}{a}$+$\frac{c}{a}$($\frac{-b}{a}$)+$\frac{c^2}{a^2}$
=$\frac{(a^2+b^2+c^2–ab–bc–ca)}{a^2}$
= $\frac{\left[\right(a+b{)}^2+(b-c{)}^2+(c-a{)}^2]}{\left(2a^2\right)}$