Question #190112

The number a is real and such that the equation x^2 +2 (a - 1) x - a + 7 = 0 has

  two different real negative solutions. One can then conclude that

  (a) a <−2; (b) 3 <a <7; (c) it is impossible; (d) none of (a) - (c).


1
Expert's answer
2021-05-07T14:23:37-0400

Given equation-

x2+2(a1)xa+7=0x^2+2(a-1)x-a+7=0


Let the required roots be -m and -n


then sum of roots mn=2(a1)    (1)-m-n=-2(a-1)~~~~-(1)


Also Product of roots m×n=a+7     (2)-m\times -n=-a+7~~~~~-(2)


m+n=2a2 and mn=7am+n=2a-2 \text{ and } mn=7-a


As, mn>07a>0a<7mn>0\Rightarrow 7-a>0\Rightarrow a<7


Also, m+n>02a2>0a>1m+n>0\Rightarrow 2a-2>0\Rightarrow a>1


So 1<a<7.1<a<7.


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