hi

[hearty flower]

Prove that the equation (9+sqrt(3)+11sqrt(2))^x + 2(5+2sqrt(6))^x - 2(sqrt(3)-sqrt(2)) = 1 has exactly one solution.

[heavy jacinth]

hail

[mild stirrup]

what do you notice about the first two terms

[hearty flower]

if the first war 9sqrt{3} i can let it to (sqrt{3}+sqrt{2})^3

but it was 9+sqrt{3}

....

[mild stirrup]

bleh didnt see the + mb

[hearty flower]

[fleet hedge]

You can transform the equation into a root-finding problem ( f(x) = 0 ), where

$$
f(x) = (9+\sqrt{3}+11\sqrt{2})^x + 2(5+2\sqrt{6})^x - 2(\sqrt{3}-\sqrt{2}) - 1.
$$

First, you prove the following statement: if a root exists, then it is unique.
This can be done by proving that ( f(x) ) is strictly monotonic on ( $\mathbb{R}$ ); that is, by showing that

$$
f'(x) > 0 \quad \text{for all } x \in \mathbb{R}.
$$

Then, to prove that a root of the equation ( f(x)=0 ) actually exists, you show that

$$
\lim_{x \to \infty} f(x) = \infty
$$

and also that

$$
\lim_{x \to -\infty} f(x) < 0.
$$

In this way, you prove that the graph of the function ( f(x) ) must cross the line ( y = 0 ) at exactly one point.

[heavy jacinth]

maabr