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Question

Solve the following equations:
(x7)(x3)(x+5)(x+1)=1680.

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Solution

(x7)(x3)(x+5)(x+1)=1680{(x7)(x+5)}{(x+1)(x3)}=1680(x2+5x7x35)(x23x+x3)=1680(x22x35)(x22x3)=1680
Put x22x=t
(t35)(t3)=1680t23t35t+105=1680t238t+1051680t238t1575=0t263t+25t1575=0t(t63)+25(t63)=0(t+25)(t63)=0t=25,63x22x=tx22x=25x22x+25=0........(i)x22x=63x22x63=0.......(ii)
Now solving (i)
x22x+25=0
using quadratic formula
x=2±44(1)(25)2=2±41002x=2±2242=1±24
Solving (ii)
x22x63=0x29x+7x63=0x(x9)+7(x9)=0(x+7)(x9)=0x=7,9
So the values of x are 7,9,1+24 and 124

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