Struggled for a bit to put the equation in a proper form but after the first few steps the solution method was apparent. Write the equation in simplest form, 3^x * 5^(2/x) = 3^2 * 5 3^(x-2) = 5^(1-2/x) Ln everything, multiply by x and distribute the ln3 and ln5. Move some stuff around and you have a quadratic in x. After simplifying we see the discriminant is a perfect square and solving further we get two solutions, x=2 and x=log_3(5)
The second solution is cool when actually plugged in. Like the immediately apparent x=2, it produces the two factors of 5 and 9, except from the other bases
I was able to formulate the quadratic, and I can guess and check the x=2 solution, but i wasn't able to simplify it when I tried to solve it. Great problem and great solution!
Correct me if I am wrong, for I have came to the solution that is much shorter and more easy to understand: 3^x * 25^(1/x) =45 3^x * 5^(2/x) = 45 3^2x * 5^2 = 45^x 3^2x *25 = 45^x 9^x *25 = 45^x 25 = 45^x / 9^x 25 = 5^x 5^2 = 5^x 2=x
Struggled for a bit to put the equation in a proper form but after the first few steps the solution method was apparent.
Write the equation in simplest form, 3^x * 5^(2/x) = 3^2 * 5
3^(x-2) = 5^(1-2/x)
Ln everything, multiply by x and distribute the ln3 and ln5. Move some stuff around and you have a quadratic in x. After simplifying we see the discriminant is a perfect square and solving further we get two solutions, x=2 and x=log_3(5)
Nice!
The second solution is cool when actually plugged in. Like the immediately apparent x=2, it produces the two factors of 5 and 9, except from the other bases
I was able to formulate the quadratic, and I can guess and check the x=2 solution, but i wasn't able to simplify it when I tried to solve it. Great problem and great solution!
Glad it helped!
Well, by "guess and check", x=2, but rigorously, what other solutions are there?
You forgot to show the graph of the functions at the end.
😊😊 I'd say the same thing.
Correct me if I am wrong, for I have came to the solution that is much shorter and more easy to understand:
3^x * 25^(1/x) =45
3^x * 5^(2/x) = 45
3^2x * 5^2 = 45^x
3^2x *25 = 45^x
9^x *25 = 45^x
25 = 45^x / 9^x
25 = 5^x
5^2 = 5^x
2=x
(3^x)^x does not equal 3^2x
Oh my bad thanks!
Another solution is here,
3^x. 25^1/x= 45
=3^x.25^1/x = 3^2×(5^2)^1/2
Compring both side we get,
X=2 😮
Near end of vid
x = 2