Roots & Logs

Algebra Level 2

If $a$ and $b$ are the two roots of the quadratic equation $x^2-3x+3=0,$ what is the value of $\log_{2}(a+b^{-1})+\log_{2}(b+a^{-1})+\log_{2}ab?$

4 5 3 2

1 solution

Siddhartha Srivastava
Mar 5, 2014

By Vieta's formulas, $ab = 3$ .

Then, $\log_2{a + b^{-1}} + \log_2{a{-1} + b} + \log_2{ab}$

$= \log_2({(a + b^{-1})(a^{-1} + b)(ab)})$

$= \log_2({(ab)^2 + 1 + 2ab})$

$= \log_2{(ab + 1)^2}$

$= \log_2{4^2}$

$= \log_2{2^4}$

$= \boxed{4}$

yes

Chanchal Ahamed - 7 years, 3 months ago

TINSON KT - 7 years, 3 months ago