If a(70a - 12b + 2c)> 0 and the quadratic equation ax^2 + bx + c = 0 has real roots alpha and beta, then which of the following statement is always false?
alpha< beta < -7
alpha < beta<= - 8
alpha > beta>0
-7 < alpha < beta < -5
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1 Answers
23let f(x) = ax2+bx+c
a(70a - 12b + 2c)> 0
\Rightarrow a( f(-5)+f(-7) -4a)>0
\Rightarrow a >0 , and \;\;f(-5)+f(-7)>4a
or\;\; [ii] a <0 , and \;\;f(-5)+f(-7)<4a
now\; a >0 , and \;\;f(-5)+f(-7)>4a >0
it is an upward opening parabola , and here also there can be cases
i.e f(-5) and f(-7) both positive , or f(-5) > f(-7) , or f(-7) > f(-5)
for case [i i]
it wil be a downward opening parabola , and there will be cases
f(-5) and f(-7) both negative , or f(-5) < - f(-7)
now try to draw all the graphs possible from the above cases