Solving A Polynomial Equation with Radicals

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[prateek.]

2:12 sir, you wrote 3root2/2 instead of 3root2/4

[broytingaravsol]

by the equation chained with the irrational number, the roots must be one with that as the multiples and considering the denominator as 4, so it can be roughly deduced as a factor of 2 in the root denominator

[moeberry]

By inspection sqrt(2)/2 is a solution and since the derivative is 3x^2 +1>or equal to 1>0 which means the function is always increasing while we have a constant on the right hand side. Therefore there is only 1 real solution and 2 complex solutions.

[pwmiles]

I used a root-finder to get the real solution, which is a fairly simple radical as I guessed. Great work as always Sybermath :-))

[pwmiles]

You know how it's gonna go. Guess a solution, then divide for the quatratic

[piman]

I looked at the equation and immediately thought to try x = (sqrt2)/2, and voila - it worked! Ten *seconds* !

[vaggelissmyrniotis]

You can write this equation as so there is one real solution x=(sqrt(2))/2 and two complex solutions

[pietergeerkens]

I got there faster (by a method I'm using more and more these days) by recognizing that
(√2⋅x)³ + (√2⋅x) - 3 = 0
after multiplying through by (√2)³ and cancelling on the right hand side. This frequently reduces, significantly, the number of possible rational roots to be hand-checked.

[realcirno]

it seems reasonable to guess that x = k * sqrt(2) where k is rational, this lets you use the rational root theorem on the resulting polynomial 8k^3 + 4k - 3 = 0

[scottleung]

I guessed and checked to find x=sqrt(2)/2 as the only real solution. Knowing this is a cubic, I then factored the original expression to get the quadratic, which gave me the other two complex solutions. Believe it or not, after all that there was no subbing necessary!

[vighnesh]

You didn't solve for complex solutions 😭

[MrLidless]

“We’re going to be solving a polynomial cubic equation”. Not quite sure why you had to say “polynomial”, not sure why having a radical complicates anything, and not sure that coming up with less than half the solutions is a win.

Normally your vids are fun and great. This was not one of them.

Also, as soon as you see that equation in a YouTube video, you *know* to substitute x=sqrt(2)y to very quickly get to the answer at the very beginning.