Brain Not Braining: Part 4 \ stacker news ~science

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333 sats \ 5 comments \ @noknees 6h science

I was stumped (but finally solved) by this seemingly simple question one of my friends challenged me, and I thought it would be good challenge for others here too :)

\sqrt{5+x} = 5 - x^2, \forall x \in \mathbb{R}

Find

x

for with proper steps/explanation for a 1000 sat bounty :)

There is a modern method to do it, but there's also a formula like the ol' quadratic formula for which I'll give extra sats if used :)

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15 sats \ 3 replies \ @Scroogey 2h

I would have squared both sides to

 $x^4 - 10x^2 - x + 20 = 0$

and then followed https://math.stackexchange.com/questions/785/general-formula-for-solving-quartic-degree-4-equations/1219804#1219804 with a=0, b=-10, c=-1, d=20. But the quadratic equation

 $v^2 + (2000-27-72*200)v + (100+12*20)^3 = 0$

has no solution, indicating that I made a mistake, and I grew tired :-)

0 sats \ 2 replies \ @noknees OP 2h

well that formula they showed is actually derived from the original quartic formula after depressing it and going case wise depending on the monic polynomial which is not the case for this one, this one is a quasi-symmetric variant, so it's not likely to give the answer :)

Good try tho! Most people won't reach till here...

0 sats \ 1 reply \ @Scroogey 2h

Oh, this is a better strategy: https://www.mathportal.org/formulas/algebra/solalgebric.php

With

a_1=0, a_2=-10, a_3=-1, a_4=20

the cubic equation becomes

 $y^3+10y^2-80y-801=0$

and with

y_1 = -9

(rational root test) the quadratic equations become

 $z^2+z-5=0$

 $z^2+z-4=0$

 $z^2-z-5=0$

 $z^2-z-4=0$

The solutions are

 $\frac{\pm 1 \pm \sqrt{21}}{2}$

and

 $\frac{\pm 1 \pm \sqrt{17}}{2}$

0 sats \ 0 replies \ @Scroogey 1h

Since squaring both sides introduced false solutions, we have to check the four solutions above by inserting them into the original equation, which shows only two of them a true solutions:

 $\frac{1-\sqrt{21}}{2}$

and

 $\frac{-1+\sqrt{17}}{2}$

0 sats \ 0 replies \ @noknees OP 2h

formula like the ol' quadratic formula

by "like" I mean "having a similar name to" not just the same