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If the equations ax^3+2bx^2+3cx+4d=0 and ax^2+bx+c=0 have a non-zero common root then prove that (c^2-2bd)(b^2-2ac)>_ 0

4 months ago

Let the non-zero common root of ax3 + 2bx2 + 3cx + 4d = 0  ..(1) and ax2 + bx + c = 0  ..(2)  be .
If  is a root of ax2 + bx + c = 0 , it will also a root of x.(ax2 + bx + c) = x.0  i.e.
ax3 + bx2 + cx = 0    ...(3).

Then  is also a root of (1) – (3) = 0   (ax3 + 2bx2 + 3cx + 4d) – (ax3 + bx2 + cx) = 0
bx2 + 2cx + 4d = 0   ...(4)

Thus, equations ax2 + bx + c = 0 and bx2 + 2cx + 4d = 0 have a common root . This implies
a2 + b + c = 0   ...(i)      b2 + 2c + 4d = 0   …(ii)

Multiply (i) by 2c and (ii) by b and then subtract (ii) from (i) to get
(2ac – b2)2 + (2c2 – 4bd) = 0      2 = 2(c2 – 2bd) / (b2 – 2ac)

is non-zero, 2 > 0,  i.e.  2(c2 – 2bd) / (b2 – 2ac)  > 0  or  2(c2 – 2bd)(b2 – 2ac) / (b2 – 2ac)2  > 0
(c2 – 2bd)(b2 – 2ac)  >  0   provided  b2  2ac.
4 months ago
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