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Question

If a,b,c are real numbers a0. If α is a root of a2x2+bx+c=0,β is a root of a2x2bxc=0 and 0<α<β, then the equation a2x2+2bx+2c=0 has a root γ that always lies between αandβ.If true write 0 else 1.

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Solution

As α is root of a2x2+bx+c=0
a2α2+bα+c=0 ----- ( 1 )

As β is root of a2x2bcc=0
a2β2bβc=0 ---- ( 2 )

Now, we have,

f(x)=a2x2+2bx+2c

f(α)=a2α2+2bα+2c

f(α)=a2α2+2(ba+c)

Using equation ( 1 ) we get,
f(α)=a2α22a2α2
f(α)a2α2
f(β)=a2β2+2bβ+2c
f(β)=a2β2+2(bβ+c)

Using equation ( 2 ) we get,
f(β)=a2β2+2(a2β2)
f(β)=3a2β2
f(α)f(β)=3a2β2a2α2, which is negative quantity.

So, one root of f(x) must lie between α and β.
α<γ<β
The answer is 0

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