# Quintic Simultaneous Equations

Solve the simultaneous equations:

$(a + b)^3 (a + b)^2 = 32b(8b^2 + 6b + 3) + b(b^3 - 3b^2 - 5b - 10) = - 8$ $a - b = 2$

Give all answers in the form $(a, b)$.

($22$ marks)

Note by Yajat Shamji
8 months, 3 weeks ago

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New solution (hence the editing of the first equation):

Leave the $= - 8$ for later, we're focusing on the first two parts of the first equation and the second equation.

First, rearrange the second equation to obtain $a = 2 + b$.

Then, substitute into $(a + b)^3(a + b)^2$.

Simplify to $(2b + 2)^5$.

Expand and simplify to $32b(b^4 + 5b^3 + 10b^2 + 10b + 5 +$$\frac{1}{b})$.

Divide all sides of the equation by $b$.

Move the $32(8b^2 + 6b + 3)$ to the LHS of the equation.

Divide the LHS by $32$.

Group like terms and you should be left with $b^4 + 5b^3 + 2b^2 + 6b + 8 +$$\frac{1}{b} =$ $b^3 - 3b^2 - 5b - 10$.

Move $b^3 - 3b^2 - 5b - 10$ to the LHS of the equation and group like terms - you should be left with $b^4 + 4b^3 + 5b^2 + 11b + 18 + \frac{1}{b} = 0$

Multiply by $b$, resulting in $b^5 + 4b^4 + 5b^3 + 11b^2 + 18b + 1 = 0$.

Set the equation equal to $- 8$ from the first simultaneous equation - you should obtain $b^5 + 4b^4 + 5b^3 + 11b^2 + 18b + 9 = 0$

Set the equation as $f(b)$.

Perform factor theorem - you'll find out that $f(- 1) = 0$, therefore $(b + 1)$ is a factor of $b^5 + 4b^4 + 5b^3 + 11b^2 + 18b + 9 = 0$

Perform the long polynomial division $\frac{b^5 + 4b^4 + 5b^3 + 11b^2 + 18b + 9}{b + 1}$

You should get $b^4 + 3b^3 + 2b^2 + 9b + 9$.

Set this equation as $f(b)$.

Perform factor theorem - you'll find out that $f(- 1) = 0$ and $f(- 3) = 0$, therefore $(b + 1)$ and $(b + 3)$ are factors of $b^4 + 3b^3 + 2b^2 + 9b + 9$

Perform the long polynomial division of $\frac{b^4 + 3b^3 + 2b^2 + 9b + 9}{(b + 1)(b + 3)}$

You should get $b^2 - b + 3$.

Now, using the discriminant of $b^2 - 4ac$, we get $- 11$, indicating imaginary solutions and the fact that this cannot be factored further.

Therefore, the factors of $b^5 + 4b^4 + 5b^3 + 11b^2 + 18b + 9$ are $(b + 1)^2(b + 3)(b^2 - b + 3)$

Now, using Brahmagupta's quadratic formula and the discriminant, we solve $b^2 - b + 3 = 0$, leading to $\frac{1 \pm \sqrt{- 11}}{2}$

This simplifies to $\frac{1}{2} \pm \frac{11i}{2}$

Therefore, the solutions for $b = - 1, - 3$ and $\frac{1}{2} \pm \frac{11i}{2}$.

Substitute all values into $a - b = 2$ and $a = 1, - 3$ and $\frac{5}{2} \pm \frac{11i}{2}$.

Therefore the solutions to the simultaneous equations:

$(a + b)^3 (a + b)^2 = 32b(8b^2 + 6b + 3) + b(b^3 - 3b^2 - 5b - 10) = - 8$

$a - b = 2$

are (in the form $(a, b)$, as requested):

$(1, - 1)$

$(- 3, - 3)$

$(\frac{5}{2} - \frac{11i}{2}, \frac{1}{2} + \frac{11i}{2})$

$(\frac{5}{2} + \frac{11i}{2}, \frac{1}{2} - \frac{11i}{2})$

Note: Only $4$ solutions are here to due to the repeated root of $(b + 1)^2$.

Test.

- 2 months, 3 weeks ago

Ok!!!

- 8 months, 3 weeks ago

Me with the help of internet.

- 8 months, 3 weeks ago

$a=\sqrt[5]{b^4+3b^3+b^2-b}-b$

- 8 months, 3 weeks ago

That's not a solution.

- 8 months, 3 weeks ago

From eq 2,

$a = b+2$

Substituting in eq 1,

$(b+b+2)^{3}(b+b+2)^{2}=b^{4}+ 3b^{3} + b^{2} - b$

$(2b+2)^{3}(2b+2)^{2}=b^{4}+ 3b^{3} + b^{2} - b$

$(2b+2)^{5}=b^{4}+ 3b^{3} + b^{2} - b$

$32(b^{5} + 5b^{4} + 10b^{3} + 10b^{2} + 5b + 1 )=b^{4}+ 3b^{3} + b^{2} - b$

$32b^{5} + 160b^{4} + 320b^{3} + 320b^{2} + 160b + 32=b^{4}+ 3b^{3} + b^{2} - b$

$32b^{5} + 159b^{4} + 317b^{3} + 319b^{2} + 161b + 32 = 0$ (I don't know how to find roots of this mammoth...)

@Yajat Shamji - This next part took a long time to calculate, mostly because I brute-forced the roots of the 5th degree term after graphing it...

One solution is A = 1 and B = -1, and other real solutions can be found when graphed -

Real Solutions -

$a = 1, b = -1$ (got it easily using Remainder Theorem)

$a = 0.464537, b = -1.53546$

$a = 1.42518, b = -0.574825$

- 8 months, 3 weeks ago

I don't get when you change $(2b + 2)^5 = b^4 + 3b^3 + b^2 - b$ into $32(b^5 + 5b^4 + 10b^3 + 10b^2 + 5b + 1) = b^4 + 3b^3 + b^2 - b$ can you explain plz.

- 8 months, 3 weeks ago

Sure.

(2b+2) = 2(b+1)

(2b+2)^5 = 2^5 x (b+1)^5 = 32 x (b+1)^5 = 32(b^5+5b^4+10b^3+10b^2+5b+1)

The last expansion is done by (b+1) x (b+1)x(b+1) x (b+1) x (b+1)

- 8 months, 3 weeks ago

I found a few complex solutions too (where $j = \sqrt{-1}$ ). I was also able to find the same real-valued solutions you did. This makes sense, since there should be five solutions for a quintic equation.

First complex solution:

$a_1 \approx 1.0708 + 0.5192 j \\ b_1 \approx -0.9292 + 0.5192 j$

Second complex solution:

$a_2 \approx 1.0708 - 0.5192 j \\ b_2 \approx -0.9292 - 0.5192 j$

- 8 months, 3 weeks ago

Nice @Steven Chase! That's all five solutions of the quintic equation. @Yajat Shamji - Challenge complete!!!

- 8 months, 3 weeks ago

Well, @Percy Jackson, @Steven Chase, you have completed the challenge, yes!

But keep it a secret for now - I want other people to solve it...

Also, I'll post my solution as well. It is done an entirely different way to yours.

And Percy, there was a simpler way than brute force (that's all I'm telling you.).

- 8 months, 3 weeks ago

Update - I have changed equation $1$ to compensate for my solution...

You have until tomorrow to solve it... again!

- 8 months, 2 weeks ago

bruh...

- 8 months, 1 week ago

- 8 months, 3 weeks ago

- 2 months, 3 weeks ago

Uhh...

Hello Yaj

Anything you'd love to say?

- 2 months, 3 weeks ago

I posted a new solution and edited it - it just took me $6$ months to figure it out...

- 2 months, 3 weeks ago

wait aren't you the one who made the question?

- 2 months, 3 weeks ago

Yes, but can't I post my solution as well?

- 2 months, 3 weeks ago

you can, but you said you took 6 months to figure out your own question.

- 2 months, 3 weeks ago

I did say that. But did anybody notice? No.

Plus, I was busy with my studies. So, not a lot of time to actually attempt to correct in these $6$ months.

I only managed to correct and figure it out a few days ago.

So, technically, it took me $45$ minutes in $6$ months...

Anyway, let's not keep arguing. Since you're here, what do you think of the solution? (since @Frisk Dreemurr won't answer.)

- 2 months, 3 weeks ago

I'd say it's not for my level, it was very detailed indeed. ( I'm pretty dumb by the way)

- 2 months, 3 weeks ago

Ok. I get detailed a lot of the time.

- 2 months, 3 weeks ago

Can you check on my new note, I really need some suggestions for my next part thx.

- 2 months, 3 weeks ago

- 2 months, 3 weeks ago

- 2 months, 3 weeks ago

Not my area of specialty (my area of specialties are number theory, primes and algebra), but I'll try my best!

- 2 months, 3 weeks ago

cool👍

- 2 months, 3 weeks ago

And as for you being dumb, use this website freely.

- 2 months, 3 weeks ago

It'll help immensely.

- 2 months, 3 weeks ago

Second to top comment. What do you think?

- 2 months, 3 weeks ago

Let me know soon!

:)

- 2 months, 3 weeks ago

Asking cuz I can see that this post is approx. six months old...

- 2 months, 3 weeks ago