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Question

Solve:
30(x2+1x2)77(x1x)12=0

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Solution

30(x2+1x2)77(x1x)12=0
x2+1x2=[(x1x)2+2]
30x2+1x2=[(x1x)2+2]7777(x1x)12=0
Let x1x=m
30[m2+2]7712=0
30m277m+48=0
30m245m32m+48=0
15m(2m3)16(2m3)=0
(2m3)(15m16)=0
2m3=0or 15m16=0
2m=3 or 15m=16
m=32 or m=1615
By resubstituting
m=x1x
x1x=32 or x1x=1615
x21=3x2 or x21=16x15
2x22=3x or 15x215=16x
2x23x2=0 or 15x216x15=0
2x24x+x2=0 or 15x225x+9x15=0
2x(x2)+1(x2)=0 or 5(3x5)+3(3x5)=0
(x2)(2x+1)=0 or (3x5)(5x+3)=0
(x2)=0 or 3x5=0
x=2 or 3x=5
and 2x+1=0 or x=53
2x=1 or x=53
2x=1 and
x=12 or 5x+3=0
5x=3
x=3/5
The roots of the given equation are 2,12,53,35

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