Q.
If x is real, then the expression x2+2x−7x2+34x−71
3445
188
Complex Numbers and Quadratic Equations
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Solution:
Let x2+2x−7x2+34x−71=m ∴x2+34x−71=mx2+2mx−7m ⇒x2(1−m)+(34−2m)x+7m−71=0
Since x is real ∴ Disc ≥ 0 ∴(34−2m)2−4(1−m)(7m−71)≥0 (17−m)2−(1−m)(7m−71)≥0 ⇒289+m2−34m−[7m−71−7m2+71m]≥0 ⇒8m2−112m+360≥0 ⇒m2−14m+45≥0 ⇒(m−9)(m−5)≥0 ⇒m>9,m>5 or m<9,m<5. ⇒m>9 or m<5 ∴m has no value between 5 and 9.