A Nice Math Olympiad Algebra Problem | A Nice Exponential Equation

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Find the value of x?
How to solve How to solve 2^x = x²

In this video, we'll show you How to Solve Math Olympiad Question A Nice Algebra Problem 2^x = x² in a clear , fast and easy way. Whether you are a student learning basics or a professtional looking to improve your skills, this video is for you. By the end of this video, you'll have a solid understanding of how to solve math olympiad exponential equations and be able to apply these skills to a variety of problems.
#matholympiad #maths #math #algebra

[sunshineslearninghub]

What is the whiteboard software that you are using to write and teach ?

[musicsubicandcebu]

7:40 . . . but aren't you assuming 4 is a solution before 'finding' it? There's also a third solution..

[robertsauvage]

En 2:06 il me semble qu'une solution x = 2 était évidente ?

[benjaminvatovez]

After finding out that 2 and 4 are solutions, we can show there is a third one:
f(x)=x².2^(-x)-1
f'(x)=2x . 2^(-x) + x². 2^(-x) . (-ln 2) >= 0 if and only if x between 0 and 2/ln 2.
(0, -1) is the minimum, (2/ln 2, 2/ln 2) is the maximum and there is an horizontal asymptot line at y=-1 for x>0. Consequently, there are 3 solutions, 1 is negative.
We can use the W function from |x|² . 2^|x|=1, x<0:
|x| . 2^(|x|/2)=1
|x| . e^(ln 2.|x|/2)=1
(ln 2.|x|/2) . e^(ln 2.|x|/2)=ln 2 / 2
ln 2.|x|/2 = W(ln 2 / 2)
x = - (2/ln 2) W(ln 2 / 2)

[AbbasGuclu-jnpo]

2^x×1/2x=x^2×1/2x
2^1/2=x^1/x
x=2 (1)
2^4=4^2
x=4 (2)

[maetalica]

How to know how many solutions are there?

[DJ.Nihad]

Have 3 solutions. That implies from continuitg

[nuhumaishanu]

😮I am confused, too many steps that look a repetition

[prollysine]

x*ln2=2*lnx, ln2/2=lnx/x, ln2*e(-ln2)=lnx*e^(-lnx), / *(-1), , -ln2=-lnx, x=2,
W(-ln2/2)=-lnx, lnx=- W(-ln2/2), x=e^(- W(-ln2/2) ), x=2, solu, x=2,

[saqarislam]

None sense, I can solve this in ten seconds