Question

# The number of positive integral values of m less than 17 for which the equation (x2+x+1)2−(m−3)(x2+x+1)+m=0,m∈R has 4 distinct real roots is

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Solution

## (x2+x+1)2−(m−3)(x2+x+1)+m=0⋯(i) Assuming t=x2+x+1 =(x+12)2+34 ⇒t∈[34,∞) f(t)=t2−(m−3)t+m=0⋯(ii) Let the roots of the equation (ii) be t1,t2 (i) For every t>34, there exists 2 distinct real roots for x2+x+1=t (ii) For every t<34, there exists no real roots For x2+x+1=t Given equation (i) will have4 distinct real roots iff both roots of equation (ii) t1,t2>34 So, the requird condition are, (i) D>0 ⇒m2−10m+9>0⇒(m−1)(m−9)>0⇒m∈(−∞,1)∪(9,∞) (ii) f(34)>0⇒916−3(m−3)4+m>0 ⇒m>−454 (iii) −b2a>34⇒(m−3)2>34⇒m>92 ∴m∈(9,∞) Hence the number of inetgral values of m less than 17 is 7.

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