German math Olympiad. A nice trick used in solving Olympiad math problem on exponential equation.

This is really an interesting video showing how simple tricks could be used to solve Olympiad problems on exponential equations.

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@begechorimaths2975

[dorienjames]

Where does it tell you that only real solutions are required?

[MARTINWERDER]

Similar solution as that one of @GyB2GyB2GyB: transform to 2^3x - 2^x = 120 .
Assume 2^X = a :
a^3 - a = 120 or a(a^2 - 1 ) = 120 or a(a - 1 ) ( a + 1 ) = 2.2.2.3.5 = 5.4.6= 120 a=5 = 2^x
x = log 5 base 2 = 2.3219
8^x = 125
8^x - 2^x = 125 - 5 = 120
Very fast solution with not much calculation needed. Calculator used for checking only.

[GyBGyBGyB]

y ^ 3 - y = 120
y * (y ^ 2 - 1) = 120
y * (y - 1) * (y + 1) = 2 * 2 * 2 * 3 * 5
(y - 1) * y * (y + 1) = (2*2) * 5 * (2*3)
(y - 1) * y * (y + 1) = 4 * 5 * 6
(y - 1) * y * (y + 1) = (5 - 1) * 5 * (5 + 1)
y = 5

[anestismoutafidis]

8^2, 33-2^2, 33=122 <=>
8^2, 322 - 2^2, 322= 120
<=> x=2, 322

[RealQinnMalloryu]

120/8÷10.40x^Log^x120/2=60 {10.40 ➖ 60}= 10.20 10.5^4 2^5.5^4 2^1.1^4 2..2^2 1.1^2 1^2 (x ➖ 2x+1) .

[kimba]

Whereas guess 'n' test is a legitimate technique, it is not very reliable, also, you failed to check y=4 without explanation.
If we factorize, we get (y-1)y(y+1)=120
That is, 3 successive integers whose product is 120. these must be close to the cube root of 120, so we see 3, 4, 5 work. y = 5.
The problem with guess 'n' test is that it can be a loong process.
Further, you should have indicated WHY you expected an integer solution, you know:
"Any cubic polynomial must have at least one real solution and if a monic polynomial with integer coefficients has real solutions they must be integers."
You can't assume the student knows this stuff.
Further, There is no indication that the solutions be real, so you MUST find the complex solutions too. (Complex, not imaginary - complex numbers have an imaginary PART). If you want only real solutions you have to say "real solutions". I mean, how hard is that?