39answer is all
the f(-x) has the sign changed only once (the x term) for all the equations
Among the following which pair of quadratic equations have the same nature of roots?
a) 13x2+41x+37 and 912+67x+13
b) 11x2+23x+25 and 59x2+45x+11
c) 31x2+14x+7 and 52x2+76x+31
d) all
Finding out the discriminant is a tedious job.... is their any other method to arrange them and know the nature of roots?
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6 Answers
39
1basic is like dis
let
f(x) = 13x^2 + 41 x + 37 \, \, \, \, and \, \, \, \, g(x)= 91 x^2 +67x + 13
now
f(-x) = 13(-x)^2 + 41 (-x) + 37 = 13 x^2 -41 x + 37
observin coeefecients of f(-x)
13 x^2 -41 x + 37
+ \,\, -\, \, \, \, +
coefficient of x^2 is 13 which is +ve
coefficient of x =-41 which is -ve
constant term = + ve
hence sign of f(-x) changes two times
from +ve to - ve and then -ve to + ve
f(x) = 13x^2 + 41 x + 37
here there is no change in signs of coefficient of x2 , x , and constant term
ie all are +ve
hence
no of positive roots = no of changes in signs of f(x) =p
no.o negative roots = no.o changes in signs of f(-x) = n
no.o complex roots = degree of f(x) - n-p =c
similarly u can continue for g(x) and other three options
1but JAI this method only tells max.no. of +ve or -ve roots so how can this b applicable.........?????
1all the roots are negative
if product is +ve then either bothe are +ve or -ve
and then see the sumif it is negative then bothe are negtive or vice versa
