If a + b = 8 and ab = 15, find the value of a4 + a2b2 + b4.
Answer :
a4 + a2b2 + b4
Above terms can be written as,
a4 + 2a2b2 + b4 – a2b2
(a2)2 + 2a2b2 + (b2)2 – (ab)2
(a2 + b2)2 – (ab)2
(a2 + b2 + ab) (a2 + b – ab)
a + b = 8, ab = 15
So, (a + b)2 = 82
a2 + 2ab + b2 = 64
a2 + 2(15) + b2 = 64
a2 + b2 + 30 = 64
By transposing,
a2 + b2 = 64 – 30
a2 + b2 = 34
Then, a4 + a2b2 + b4
= (a2 + b2 + ab) (a2 + b2 – ab)
= (34 + 15) (34 – 15)
= 49 × 19
= 931