Math Olympiad Question | Many SKIP This Critical Step

Be careful! A tricky algebra question. What do you think about this problem?
If you're reading this ❤️
Hello My Friend ! Welcome to my channel. Have a Great Day:)
‪@higher_mathematics‬
#maths #algebra

[@stabbysmurf]

Just by imagining the graph of a^4 and the graph of (a-1)^4 (the same shape shifted right by 1,) you know they cross at 1/2. You can then factor (a-1/2) from (a^4 - (a-1)^4) and get a quadratic with the other two roots.

[@AlcyonEldara]

x^4 = 1 has 4 complex roots: 1; -1; i and -i
x^4 = k has 4 complex roots, the 4th root of k multiplied by the roots of the previous equation
So a^4 = (a-1)^4 has 4 possible equations for the roots:
a = 1.(a-1) impossible
a = -1.(a-1) a = 1/2
a = i.(a-1) a = -i/(1-i) = (1-i)/2
a = -i.(a-1) a = i/(1+i) = (1+i)/2

[@bjornfeuerbacher5514]

Thanks, that's how I would have solved it, too. Much quicker and easier than in the video!

[@rbenmass]

great math video. super!!

[@Jon60987]

Nice problem, nice solution :)

[@lornacy]

Nice!

[@janospeszeki3288]

good job

[@adekunlesunday5897]

Where is the fourth solution

[@silver6054]

Reusing "a" for the difference of two squares isn't a great choice! But otherwise interesting approach

[@philpollack8140]

I was going to comment this as well - totally agree. A beginner can easily be confused by the re-use of "a" here.

[@waliaphellps1745]

Great! But I didn’t understand why taking 4th root leads to nonsense (a-a=-1). At least in principle strikes to me as a correct approach to do the same thing in both sides of the equation.

[@MightyBiffer]

the problem lies in that he forgot to use the absolute value function. For example, if a squared equals b squared that does not mean that a = b. It means that |a| = |b|. If a = 2 then a^2 = 4. Since b^2 also equals 4 and an obvious possible value of b is 2. But -2 also satisfies the equation,

[@chitlitlah]

MightyBiffer is right. You could take the square root or the fourth root of both sides, but you need to put a +/- on one of those sides. If you put it on the left side, you get a = a - 1 and -a = a - 1. The first one obviously has no real answer, but the second one can easily be worked out to a = 1/2. I don't think you could work out the complex solutions that way though.

[@ScientistPrepper]

@@MightyBiffer Nice explanation.

[@iutu8235]

and the 4th solution?

[@Jon60987]

No 4th root. If you multiply out (a-1) to the 4th you get a to the 4th plus a cubic polynomial. The a to the 4th terms subtract off and cancel each other out leaving a cubic polynomial (3 solutions).

[@comdo777]

asnwer=a1 isit

[@bjornfeuerbacher5514]

Say, do you write nonsensical comments on _every_ math video? :D