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If-4a-c-2-4b-2-then-one-root-of-ax-2-bx-c-0-lies-in-1-2-2-2-1-1-3-2-4-2-

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Question Number 20053 by Tinkutara last updated on 21/Aug/17
If (4a + c)^2 ≤ 4b^2 then one root of ax^2 + bx + c = 0 lies in (1) (−2, 2) (2) (−1, 1) (3) (−∞, −2) (4) (2, ∞)
If(4a+c)2⩽4b2thenonerootofax2+bx+c=0liesin(1)(−2,2)(2)(−1,1)(3)(−∞,−2)(4)(2,∞)
Answered by ajfour last updated on 21/Aug/17
(4a+c)^2 −4b^2 ≤ 0 ⇒ (4a+2b+c)(4a−2b+c) ≤0 ⇒ f(2)f(−2)≤0 ⇒ f(x) changes sign or is zero in [−2,2], hence one root lies in [−2, 2].
(4a+c)2−4b2⩽0⇒(4a+2b+c)(4a−2b+c)⩽0⇒f(2)f(−2)⩽0⇒f(x)changessignoriszeroin[−2,2],henceonerootliesin[−2,2].
Commented by Tinkutara last updated on 21/Aug/17
Thank you very much Sir!
ThankyouverymuchSir!

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