If $a^2 + 4b^2 + 25c^2 + 18 = 2(a - 2b + 20 c),$ then what is the value of $(a + 2b + 5c)$ ?
Answer & explanation
Correct answer: option 2
a2 + 4b2 + 25c2 + 18 = 2(a - 2b + 20c)
= a2 + 4b2 + 25c2 - 2a + 4b - 40c + 18 = 0
= (a2 - 2a + 1) + (4b2 + 4b + 1) + (25c2 - 40c + 16) = 0
= (a - 1)2 + (2b + 1)2 + (5c - 4)2 = 0
a - 1 = 0, a = 1
2b + 1 = 0, b = \(\frac{-1}{2}\)
5c - 4 = 0 , c = \(\frac{4}{5}\)
so the value of $(a + 2b + 5c)$ = $(1 - 2 × \frac{1}{2} + 5 × \frac{4}{5}$) = 4