JEE Main 2016 · Mathematics · Quadratic Equations · hard

The number of integer values of m for which both roots of x^2-(m+1)x+(m^2-2)=0 are real and positive is

Single correct option

The number of integer values of

m

for which both roots of

x^2-(m+1)x+(m^2-2)=0

are real and positive is

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Answer: (B) $1$

Three conditions are needed. Real roots:

D=(m+1)^2-4(m^2-2)=-3m^2+2m+9\ge0

, so

\frac{1-2\sqrt7}{3}\le m\le\frac{1+2\sqrt7}{3}

, i.e. approximately

-1.43\le m\le2.10

. Positive sum:

m+1>0

, so

m>-1

. Positive product:

m^2-2>0

, so

|m|>\sqrt2

. Combining,

m>-1

and

|m|>1.414

force

m>1.414

, and with

m\le2.10

the only integer is

m=2

. Check:

x^2-3x+2=0

has roots

1,2

. Hence exactly one value.

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