Let-a-b-c-R-a-0-such-that-a-and-4a-3b-2c-have-the-same-sign-Show-that-the-equation-ax-2-bx-c-0-can-not-have-both-roots-in-the-interval-1-2-

Question Number 18384 by Tinkutara last updated on 19/Jul/17

Let a, b, c \in R, a \neq 0, such that a and

4 a + 3 b + 2 c have the same sign . Show

that the equation {a x}^{2} + b x + c = 0 can

not have both roots in the interval

(1, 2) .

Answered by Tinkutara last updated on 22/Jul/17

Let α and β be the roots of the equation .

Let a ⩾ 0 s o t h a t 4 a + 3 b + 2 c ⩾ 0

\Rightarrow 4 + 3 \frac{b}{a} + 2 \frac{c}{a} ⩾ 0

\Rightarrow 4 - 3 (α + β) + 2 α β ⩾ 0

\Rightarrow (α - 1) (β - 2) + (α - 2) (β - 1) ⩾ 0,

which is not possible if both α and β

belong to (1, 2) .