problem with two squared variables

[12obin]

I could've sworn there was a spot for homework help but I can't find it so I hope this is an okay place.

I have this problem for an assignment that's like, a "if this is true, then prove that this other thing is also true" one and I kind of don't want to give the whole thing here because I want to solve it myself still, but I'm pretty sure the first step should involve isolating a variable in the first equation so I can express a in terms of b or vice versa.

the first equation is

a² + b² = 23ab

more rambling:

Spoiler:

and I feel vaguely like there's a factoring trick I'm forgetting for a situation like this but I just can't manage it. In my frustration I sort of brute-forced a solution honestly, where one variable is 1 and the other is ~22.96, but that's not actually very helpful. I was hoping that if I found a solution, it would give me a clue on how to isolate a variable, but nah. and I tried graphing it on Desmos but it sort of just...wouldn't? and I can't blame it.

Anyway maybe I'm coming at it all wrong. The second equation, that I'm supposed to prove is true, is a log one fwiw, and I can post it later if needed but I don't have it in front of me right now regardless.

kthx

[ucim]

What is (a+b)² ?

Jose

[12obin]

What is (a+b)² ?

Jose

a² + 2ab + b²

am i misunderstanding the question?

[ucim]

It's a hint. Take a look at your original equation and see if you recognize parts of it there, where you can substitute them. That may simplify the equation.

Also, look for symmetries - a set of solutions where a=b. (Check also for a = -b, since (-b)² = b²) Note - there may not be any.

Then look for solutions where a=kb. Simply substitute kb for a, reducing the number of variables (k is a constant). If there are solutions of that form, you should be able to figure out what k must be.

Then look for solutions where a=0 or b=0.

Hope that gets you started.

Jose

[Derek]

If you want to express a in terms of b then,

Spoiler:

Complete the square.

a^2 - 23ab = -b^2

Start from here and complete the square on the left side, then solve for a.

[Eebster the Great]

Have you already learned how to solve quadratic equations with constant coefficients? Because this is pretty much the same thing.

[12obin]

it turns out i came at it all wrong and i didn't need to isolate a variable. if i played with the other equation, i would have ended up with a² + b² on one side. it's fine though, it was a group assignment and i solved other ones and other people solved this one in the end.

still, thanks for reminding me about completing the square, a thing i had completely forgotten was possible.