Simple solution to a complicated problem

Algebra Level 3

Natural numbers $a$ abd $b$ are such that $a>1$ and $b=a+1$ . Let $\alpha$ and $\beta$ be the roots of the equation below.

$x^2-\left(3+a^{\sqrt {\log_ab}}-b^{\sqrt {\log_ba}}\right)x-2\left(b^{\log_ba}-a^{\log_ab}\right)=0$

What is the value of $\dfrac \alpha \beta +\dfrac \beta \alpha$ ?

The answer is 2.5.

1 solution

A Former Brilliant Member
Oct 18, 2019

Let $log_ab=x$ . Then $a^x=b$ , or $a=b^{\dfrac{1}{x}}$ , $log_ba=\dfrac{1}{x}$ . Therefore $b^{log_ba}=a$ , $a^{log_ab}=b=a+1$ , and $a^{\sqrt {log_ab}}=b^{\sqrt {log_ba}}$ . Hence the given equation reduces to the form $x^2-3x+2=0$ , whose roots are $α=2,β=1$ . Therefore $\dfrac{α}{β}+\dfrac{β}{α}=2+\dfrac{1}{2}=\boxed {2.5}$

Simple solution to a complicated problem

The answer is 2.5.

1 solution

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