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solving equations but they get increasingly more impossible?

blackpenredpen

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We will solve 4 impossible-looking equations, sqrt(x)+sqrt(-x)=2, ln(x)+ln(-x)=0, e^x+e^(-x)=0, and sin(x)+sin(-x)=2. From verifying the domains of the functions to finding real solutions, we go step-by-step through each equation. And just when you think you've got it figured out, we hit you with one more equation that's sure to blow your mind! Get ready to flex your mental muscles and solve some of the trickiest equations out there. #math #equation #blackpenredpen

0:00 sqrt(x)+sqrt(-x)=2
3:27 ln(x)+ln(-x)=0
6:10 e^x+e^(-x)=0
10:29 sin(x)+sin(-x)=2
Here's the video for Q4 Video

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20 Comments

@oreocookiedough 4 years ago

Putting these no solution questions as a pre-cal bonus question saying "graph the function" would be humorous when the people who don't know what to do leave it blank and then you mark it correct.

@lolosharacarrot5507 11 months ago

2:30 diabolical sentence

@ecg3703 4 years ago

The last one has no solution at all because sin(-x) is equal to -sin x so the equation sin x+sin(-x)=2 is the same as saying sin (x)-sin(x)=2 now we can cancle the two sines and we get the equation 0=2 and it has no solution.

@S.XKYEDITS 3 years ago

There neither real nor imaginary solution because for any x, sinx+sin(-x) =sinx-sinx=0

@ffggddss 4 years ago (edited)

Here's an alternate take for the exponential equation.
That parabola-like curve is a catenary; and eˣ + e⁻ˣ = 2cosh(x); twice the hyperbolic cosine. But from Euler's formula, (circular) cosine can be written
cos(x) = ½(e ͥˣ + e⁻ ͥˣ) = cosh(ix); likewise, because cos and cosh are even functions,
cosh(x) = cos(–ix) = cos(ix)

So eˣ + e⁻ˣ = 2cosh(x) = 0, means that
2cos(ix) = 0 = 2cos(–ix)
But we know where cosine is 0:
–ix = (n+½)π ; x = (n+½)iπ

And that's another way to solve this one.

Fred

PS. Great idea, this set of problems!

@Paul-222 3 years ago

For #3, it’s interesting that if you substitute e^i(pi) = -1 too soon or too late, you get stuck with a tautology where x = anything.

@maxwellcody6457 4 years ago

It looks like sin(x)+sin(-x) = 2 has no solutions since sin(x) is an odd function, even with using Euler's formula it leads to a contradiction of 0=2.

@justushinkelmann8020 4 years ago

Holding the poceball gives him the ultimate mathematical power

@justintroyka8855 4 years ago (edited)

If you, like me, are bothered by the fact that the process in part 1 doesn't yield both solutions, then read on. The reason is because a number has two square roots, and complex numbers don't have a preferred one of the two like positive real numbers do - you can't just take the "positive" square root because most complex numbers don't have a purely real, positive square root. More specifically, the problem is in the step √[-x] = i√x. Check this out:
√[-(-x)] = i√[-x] = i*i*√x = -√x.
So √x = -√x, which is absurd. The way to resolve this is that √[-x] may be either i√x or -i√x, depending on which square root of a number is being chosen.

@kepler4192 4 years ago

7:47 you said it yourself it's literally a cosh function, exactly 2cosh(x) 😆

@drpeyam 4 years ago

This is unreal 😂

@miscccc 4 years ago

Petition to make I dont like to be on the bottom, I like to be on the top merch.

@xavierwainwright8799 4 years ago

sin(x)+sin(-x)=2 <=> sin(x)-sin(x)=2 <=> 0=2. There are no solutions.

@luvHugeBressts 4 years ago

İ can do that! İn the last question, it have no solution. Because if we try to do that, we get 0=2 and that was an issue.

@antonyzhilin 4 years ago

I have no idea where to start with the last problem, it's so odd tbh

@prajwalpai7604 2 years ago

For the first equation we can square both sides twice to get a quadratic equation when we solve it we'll get ±2i directly

@erichsu3325 3 years ago (edited)

Junior Math: 1+1
Highschool Math: ax^3+bx^2+xc+d
University Math: I don't know anymore. - Ho Lee Fuk

@stpat7614 2 months ago

Mistake at 1:52. √(-x) = i * √(x) only if x is a non-negative real number. For instance, if x = -2, then √(-[-2]) ≠ i * √(2).

@mrgreenskypiano 4 years ago

This is honestly one of my favorite videos of yours - it’s very clear and concise but still enough content to fill a 10-minute video! All of these sections are different but have the same theme so it still feels like one video - my rating is (pi^2 + 1)/10