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If |b|>|a+c|,then show that exactly one root of the equation ax^2+bx+c=0 lies in the interval (-1,1).

If |b|>|a+c|,then show that exactly one root of the equation ax^2+bx+c=0 lies in the interval (-1,1).

Grade:upto college level

1 Answers

Radhika Batra
247 Points
7 years ago
square |b|>|a+c| 

we get b^2 > (a+c)^2
b^2-(a+c)^2>0
(b+a+c)(b-a-c)>0
we get b>a+c  and b<-(a+c)
or 0>a+c-b   ,     a+b+c<0
f(1) = a+b+c   f(-1) = a + c - b
f(1)<0     f(-1)>0

so graphically the quadratic equation has to cut the x axis at any point b/w -1 and 1

so proved that there lies only one root b/w -1 and 1

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