Question

O valor do log (a+b), sabendo que a e b são as raízes da equação x² - 100x -1100 = 0, é:
a) -2
b) 2
c) 1
d) -1

Answer 1

Utilizando Bhaskara, vamos começar achando as raízes da equação:

$\Delta~=~(-100)^2-4\cdot1\cdot(-1100)\\\\\Delta~=~10000~+~4400\\\\\boxed{\Delta~=~14400}\\\\\\a~=~\dfrac{100+\sqrt{14400}}{2\cdot1}~=~\dfrac{100+120}{2}~=~\dfrac{220}{2}~~~\rightarrow~~\boxed{a~=~110}\\\\\\b~=~\dfrac{100-\sqrt{14400}}{2\cdot1}~=~\dfrac{100-120}{2}~=~\dfrac{-20}{2}~~~\rightarrow~~\boxed{b~=\,-10}$

Agora sim, calculando o logaritmo:

$\log\,(a+b)~=~x\\\\\\\log\,(110+(-10))~=~x\\\\\\\log100~=~x\\\\\\100~=~10^x\\\\\\10^2~=~10^x\\\\\\10\!\!\!\!\backslash^2~=~10\!\!\!\!\backslash^x\\\\\\\boxed{x~=~2}~~\rightarrow~~Letra~b$