Harvard Entrance Exam - Algebra - Very Advanced Difficulty

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2^x + x = 20
This problem tests knowledge of advanced algebraic concepts, particularly the application of the Lambert W function, which is used to solve equations where the unknown variable appears in both an exponent and a linear term.
• Competition Level: International Mathematical Olympiad (IMO) or higher-level university entrance exams like MIT or Harvard entrance exams.
• Grade Level: 12th grade or early undergraduate.
• Difficulty: Very Advanced, due to the involvement of the transcendental Lambert W function and its application in solving non-trivial equations.
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[@DriftinVr]

2^x=20-x
2^20=(20-x)*2^(20-x)
2^4*2^16=(20-x)*2^(20-x)
16*2^16=(20-x)*2^(20-x)
20-x=16
x=4
boom easier method no need for e or natural logarithms

[@DriftinVr]

this uses tetration

[@mathXanswer]

love this answer, thanks for sharing!

[@atreidesson]

so, you use the fact that x*2^x is increasing and continuous so it's a bijection. But you could do that with x+2^x equals 4+2^4 as well, only that requires guessing. Also the algebra here seems to be less important and less advanced than actually explaining why the function is a bijection.

[@nskies2001]

Me, 5 seconds in the video:
"That's a f__ing 4 !" :D

[@EffigyOfficial]

I feel like should've automatically failed if you needed algebra to solve this

[@ariamh823]

Fun fact: in some exams guessing the answer based of some sore of principal is accepted. You have to say that you got it by guessing and show some proof; so it doesn't count as cheating.
Here it would be:
2^x = 20-x
I deduced that x is a positive number. I put simple numbers that would make sense and 4 was the answer

[@victorsladkovsky5653]

By taking the derivative, you can see that the function in the left hand side is growing and is equal to a constant. That means that the equation has only one solution which is easy to guess, it's four.

[@MelodySparkleroni490]

this is taking a very simple question and providing a very convoluted, unnecessarily long answer... you can go from point a to b, but this video works backwards and goes to point z, y, x, and eventually finds b.

[@mathXanswer]

You are not wrong, what's your solution tho?

[@alchgaming7585]

@@mathXanswer use a calculator :)

[@lipmajesty383]

I just solved it in my mind.

[@jasonmonks8423]

I literally just worked it out in my head lol
Its 4

[@dutchhondarebel]

I suck at math/algebra, but even I can figure out within 5 seconds that the answer is 4 by just looking at the initial equation.

[@kolega881]

you can use logic to solve this, x must be larger than 0, x must be lower than 5, or else 2^5=32, x must be an even number or the answer would be odd, and provided those limits, you only have 2 possible answers, 2 and 4, plug in both and voila, 4 is the answer
assumptions
x>0
x 2^2+2 =/= 20
for x=4 -> 2^4+4 = 20
x=4 is the solution (this method only works if x is a whole number, which in this case it is)

[@SaintUpthrust]

It works here, but generally there no guarantee that x is a whole number.

[@bosssnurp5912]

I knew it was 4 in 3 seconds

[@tmcal77]

Step 1: Notice that 𝑥 = 4 is *a* solution to 2^𝑥 + 𝑥 = 20.
In other words, putting 𝑓(𝑥) = 2^𝑥 + 𝑥 − 20, we have a solution 𝑥 = 4 to the equation 𝑓(𝑥) = 0. It remains only to check whether there are any other solutions.
Step 2: Compute the derivative of 𝑓(𝑥) to see that 𝑓(𝑥) is increasing for all real 𝑥.
In other words, the graph 𝑦 = 𝑓(𝑥) can cross the 𝑥-axis only once. We already know that it crosses at 𝑥 = 4, so that must be the only value of 𝑥 at which the graph 𝑦 = 𝑓(𝑥) crosses the 𝑥 axis.
Therefore, 𝑥 = 4 is indeed the only solution to 𝑓(𝑥) = 0. That is, 𝑥 = 4 is the only solution to the original equation.