1dint get
if a< b< c< d, then prove that for any real α,the quadratic eqn (x-a)(x-c)+α(x-b)(x-d)=0
has real roots.
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2 Answers
13Arrey dekh bhai....
f(x)=0, is a quadratic, can have 2 roots
(v have to prove tht both of them are real, i.e. the graph of y=f(x) really cuts the x-axis (whether cuts at 2 distinct points, or a single point isn't a matter of concern, coz there can't be a single imaginary root.)
f(a) = α(a-b)(a-d) > 0 (Assuming α +ve)
Also, f(b) < 0. => tht f(x) must be 0 at some pt. b/w x=a & x=b.
Same way f(d)>0 & f(c)<0, so another real root lies b/w x=c & x=d.
Hence both the roots are real...
Note tht even if "α" is negative, the result won't change.
btw akari's explanation was more-or-less complete but since....i hope he won't bash me fr tht [3]