Haha, not everyone has that same quick intuition, so using algebra helps break it down for those who need to work through the steps. It’s more about mastering the method than just finding the answer, and sometimes those steps can be a good learning moment!
@@mathXanswer You have a valid point. I was coming from an engineering perspective and the best method for me for a problem like this seemed to be to try the most obvious values. Even if you don't get it right that way, you get an interval where you know where the solution will be and that might help you with the solution
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
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.
I wondered if there are complex solutions. I asked open ai. It replied: The primary real solution to the equation is 𝑥 = 4. However, complex solutions likely exist, involving oscillations governed by ln 2. A full solution would require numerical methods (e.g., Newton's method) to find specific complex pairs that satisfy the real and imaginary part equations simultaneously. I asked it to try using Newton's method and it replied: Using Newton's method with an initial guess of 𝑥0 =3 + 1 i , we converged to the solution: x = 4.000000004 + an imaginary part of less than 10^-9 i in 5 iterations. From this, I am guessing that the only answer (real or complex) is x = 4. Phew!
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.
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.
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)
I get where you're coming from! The equation does seem simple at first glance, but solving it in a general mathematical way requires working through the equation methodically and it might have more than one answer. Glad you found the answer quickly though! Sometimes visual intuition works faster than algebra.
@@mathXanswer At 5:40 in the video, when you split 2^20=2^4*2^16, which is 2^4*2^2^4=2^(4+2^4), aren't you just doing the same as the rest of us when we spotted that 20=4+2^4 (without the complications of using the W Lambert function and raising 2 to the power of the equation we want to solve)? What led you to split 2^20 that way?
@@andrewnewman991 Not sure if I understood your point, but we applied the W function in the previous step on each side of the equation to get the right side of the equation easier to solve and after that we are stuck with W function, so the process was for getting rid of the W on the left side. Hope this helps..
@@mathXanswer My point is that the only reason you're breaking 2^20 into two parts (2^4)*(2^16) is because you know that 2^4, the first expression, is equal to 16, the index in the second expression, i.e. you know that 20 = 4 + 2^4, i.e. you already know the answer to the question!
@@andrewnewman991 I see, you are not wrong that's basically "the question" of guessing at the first place, the goal of this video is mostly playing with W function in case the answer is not real numbers, which is not the case here!
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.
We can make step 2 slightly quicker by expressing the problem as 2^x =20-x. We know that 2^x is a strictly increasing function of x and that 20-x is a strictly decreasing function of x, so they can only cross once. It avoids having to compute the derivative.
I feel like should've automatically failed if you needed algebra to solve this
Haha, not everyone has that same quick intuition, so using algebra helps break it down for those who need to work through the steps. It’s more about mastering the method than just finding the answer, and sometimes those steps can be a good learning moment!
@@mathXanswer You have a valid point. I was coming from an engineering perspective and the best method for me for a problem like this seemed to be to try the most obvious values. Even if you don't get it right that way, you get an interval where you know where the solution will be and that might help you with the solution
Me, 5 seconds in the video:
"That's a f__ing 4 !" :D
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
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.
That's the way to show it has only one answer!
I wondered if there are complex solutions. I asked open ai.
It replied:
The primary real solution to the equation is 𝑥 = 4.
However, complex solutions likely exist, involving oscillations governed by
ln 2. A full solution would require numerical methods (e.g., Newton's method) to find specific complex pairs
that satisfy the real and imaginary part equations simultaneously.
I asked it to try using Newton's method and it replied:
Using Newton's method with an initial guess of 𝑥0 =3 + 1 i , we converged to the solution:
x = 4.000000004 + an imaginary part of less than 10^-9 i
in 5 iterations.
From this, I am guessing that the only answer (real or complex) is x = 4.
Phew!
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.
You are not wrong, what's your solution tho?
@@mathXanswer use a calculator :)
I just solved it in my mind.
I literally just worked it out in my head lol
Its 4
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
this uses tetration
love this answer, thanks for sharing!
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.
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)
It works here, but generally there no guarantee that x is a whole number.
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.
I get where you're coming from! The equation does seem simple at first glance, but solving it in a general mathematical way requires working through the equation methodically and it might have more than one answer. Glad you found the answer quickly though! Sometimes visual intuition works faster than algebra.
@@mathXanswer At 5:40 in the video, when you split 2^20=2^4*2^16, which is 2^4*2^2^4=2^(4+2^4), aren't you just doing the same as the rest of us when we spotted that 20=4+2^4 (without the complications of using the W Lambert function and raising 2 to the power of the equation we want to solve)? What led you to split 2^20 that way?
@@andrewnewman991 Not sure if I understood your point, but we applied the W function in the previous step on each side of the equation to get the right side of the equation easier to solve and after that we are stuck with W function, so the process was for getting rid of the W on the left side. Hope this helps..
@@mathXanswer My point is that the only reason you're breaking 2^20 into two parts (2^4)*(2^16) is because you know that 2^4, the first expression, is equal to 16, the index in the second expression, i.e. you know that 20 = 4 + 2^4, i.e. you already know the answer to the question!
@@andrewnewman991 I see, you are not wrong that's basically "the question" of guessing at the first place, the goal of this video is mostly playing with W function in case the answer is not real numbers, which is not the case here!
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.
We can make step 2 slightly quicker by expressing the problem as 2^x =20-x. We know that 2^x is a strictly increasing function of x and that 20-x is a strictly decreasing function of x, so they can only cross once. It avoids having to compute the derivative.
holy heck they did it again
X=4 mental rith
I knew it was 4 in 3 seconds