Question

if 3a - 2b + 5c =5 and 6ab + 10bc - 15ac = 14, find the value of 27a^3 + 125c^3 + 90abc -8b^3

Answer 1

3a-2b+5c=5
Squaring both the sides
9a²+4b²+25c²-12ab-20bc+30ac=25
9a²+4b²+25c²-2(6ab+10bc-15ac)=25
9a²+4b²+25c²-2(14)=25
9a²+4b²+25c²=53
(3a)²+(-2b)²+(5c)²=53------------------(1)
Now to find:
27a³+125c³+90abc-8b³
=(3a)³+(-2b)³+(5c)³+3(3a)(-2b)(5c)
=[(3a-2b+5c){(3a)²+(-2b)²+(5c)²-12ab-20bc+30ca}]
=5[{(3a)²+(-2b)²+(5c)²-2(6ab+10bc-15ac)}]
=5[{53-2(14)}]
=5(25)
=125