How to solve this quadratic equation? Is it "all real numbers" or "no solution"? Reddit r/maths

What is the answer to the quadratic equation 5-x^2=1-(x+2)(x-2)? This is not a usual quadratic equation because you will notice the terms will cancel out. Be sure we draw the correct conclusion. Subscribe to ‪@bprpmathbasics‬ for more algebra tutorials.
Original post on Reddit r/maths: / tlaybcf6bj
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[@edwardpacman7082]

The first case: "No matter what x is, the equation is always true." The second case: "No matter what x is, the equation is never true."

[@WilliamBeason]

The only way I can imagine the equation being false is if you're dealing with a non-distributive algebra. In which case, like, why, but also, like, why didn't you know this before starting the problem.

[@tobybartels8426]

If you're sure that it's either always true or never true, then you can just try one number and see if it's true.

[@rdspam]

If the answer has to be one of the two, just set x=0, see that 5=5, then it can’t be “no solution”

[@HD-fy2wu]

When you have f(x)=f(x), it's true for all x, be it real number or complex number. As long as f is a well defined function, it should only have a unique output. Let f(x) = 5-x², then when you reach 5-x² = 5-x², you can already conclude all solutions for x.

[@xwtek3505]

You also have to make sure that all the previous steps are reverse implication, though. (i.e. if you have f(x) =g(x) -> h(x) =h(x), it's not true that f(x) =g(x) for all x)

[@hafizusamabhutta]

Can't it be true for complex numbers also?

[@taito404]

I'm just gonna reply here to get a notification for someone who has the answer

[@chaoticoli09]

Yes, all identities for real numbers are also identities for complex numbers because everything vanishes when solving and you retain the primary properties of reals.

[@cmdion]

Yes, this extends to all complex numbers.

[@xinpingdonohoe3978]

Complex numbers, like real numbers, are both commutative and associative. It is certainly true that things like this will also hold for complex numbers. So will things like sin²(z)+cos²(z)=1, even if -1≤sin(z)≤1 is no longer true.

[@seanhunter111]

Yes. Try it where x=sqrt(-1). LHS=5--1=6. RHS = 1-(-1-4)=6. So x = sqrt(-1) satisfies the equation. That isn't surprising because the lefthand side and righthand sides are literally identical. You might as well write "x=x" as write the equation given.

[@BleuSquid]

Why is it only "all real numbers"? This holds true for any complex number as well.

[@andreasnesse04]

I guess he didn’t include it in his answer because complex numbers wouldn’t be introduced yet at this degree of math and It might confuse students at this level instead of making them understand the basics.

[@MK-13337]

Since the equation is a tautology (the equation reads 5=5) it works for anything you put into x that follows the rules of arithmetic. So x could be a function if you wanted.

[@jb7650]

Why is it only "all complex numbers"? This holds true for quaternions as well.

[@nordicexile7378]

@@jb7650 THIS one may, but given that some number systems lose commutativity under multiplication you have to be careful that none of the intermediate steps violate that condition. Same is true in regular algebra... for example if you have something like: (3x-3)/(x-1) = 3 If you cross multiply, you end up with the tautology 3x - 3 = 3x - 3 which looks like "all solutions" when really x = 1 is NOT a solution of the original expression!

[@devooko]

​@@nordicexile7378wow amazing math

[@rainerzufall42]

3:00 "Ans: all real numbers"... Why not "Ans: all numbers (complex or real)"? When you had no real solution, you would also look for the complex solutions. And the Fundamental Theorem tells you, there are always two complex solutions (distinct or not) for a != 0. Unless explicitly limited to real solutions, I would always expect all the 2 solutions from IC.

[@aMartianSpy]

tautology

[@davidlloyd1526]

"This equation says nothing about the value of X"

[@justsaadunoyeah1234]

This equation does say something about the value of x

[@The_Commandblock]

​@@justsaadunoyeah1234No, ist just like saying x=x

[@justsaadunoyeah1234]

@@The_Commandblock yeah that says something about x

[@The_Commandblock]

@@justsaadunoyeah1234 It tells you that x is =x, wow. Thats litterally true for any number, x=x is useless in a system of equations. Its more of an identity just like e^ix = cos(x) + isin(x), we know that thats true for any amount of x but it doesnt Tell us anything about the value of x

[@justsaadunoyeah1234]

@@The_Commandblock bruh x=x tells us something about x. It tells us that x belongs to the set of... well... things

[@Bodyknock]

I’m still wondering why the person who posted that question thought there might not be any solutions? 🤷‍♂️

[@wobaguk]

I think on the grounds that the x's cancel out, so on the face of it there isnt an 'x' left to be equal to anything?

[@ethanebang8902]

Ain’t this basically x=x?

[@japanpanda2179]

Correct

[@goseigentwitch3105]

there's no need to stop at only real solutions either you could use quaternions if you like

[@ytsimontng]

You could write it as 0×x^2=0 for intuition

[@JakubS]

True for all X part of the real numbers

[@qtpi0]

im sorry but is this doable? integral((sqrt(1-(lnx)²))/(lnx))dx

[@random22453]

Nope, this cannot be integrated using elementary functions

[@xinpingdonohoe3978]

Not only is it not elementary, I'm pretty sure the standard special integral functions, like erf(x), li(x), Ei(x), etc., won't be able to deal with it.

[@qtpi0]

@@xinpingdonohoe3978 what about hypergeometric? any idea? i tried with it for a bit and reached integral((e^x)/(xsqrt(1-x²)))dx

[@random22453]

@@qtpi0 nah im pretty sure its not possible with known mathematical functions

[@random22453]

@@qtpi0 yep wolframalpha also says that no standard functions exist for this

[@DARKi701]

for the second case, you could also have it explained by geometry

[@neverg0nnag1vey0uup]

Y'all didn't learn this in high school?

[@inyomansetiasa]

Hello

[@bentpc]

Much ado about nothing!Just sketch the graphs to observe.

[@memebaltan]

first

[@AwesomeCamera87_HD]

dude the video itself came out only 2 minutes ago how you commented 3 months ago 💀💀💀

[@bigbrewer3375]

@@AwesomeCamera87_HD maybe he had early access to the video

[@lumina_]

wtf

[@AizenSosukesama]

Actually first undesputedly

[@random22453]

Blud time travelling

[@GabriTell]

This is basic pedicate logic tbh 💀

[@Sinsults]

Not gonna watch the video. X=0

[@Sinsults]

@@a_man80 Nah, I already closed the video without watching. It's illegal for me to click on the video again.

[@adamkarolak3544]

Bad trolling is good! P.s just to be sure that someone read this comment will not take this seriously. X=0 would be actual solution, but in case of this video it's only one of infinitely many solutions.

[@hallrules]

Technically ur right