High school algebra question

[hushed sage]

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[cold jolt]

I was planning to isolate the radical and then square both sides but my teacher says to avoid high degree polynomials

So im not sure where to start now

[topaz pelican]

x^2+2x+24 should be positive real number because it's in square root

[topaz pelican]

U will get high powers but u will get values eventually

[cold jolt]

Yeah I got degree 4

But the course im taking is a Problem Solving course so he said its not what he wants us to do

[cold jolt]

Yeah, but im not sure what I can do with the quadratic in sqrt 😭

[iron herald]

x^2+2x appears twice !!

hmm

[cold jolt]

O

Substitution ?

[iron herald]

that is what i thought yea

[cold jolt]

I will try that tysm

[topaz pelican]

Leave that sqrt aside for a while ig we can make like squares instead

x^2+2x-41

More like x^2+2x+4

[cold jolt]

Huh

Wait which square is that

[topaz pelican]

x^2+2x+4-45=(x+2)^2-45

[cold jolt]

thats not

x^2 + 4x + 4

= (x+2)^2

[iron herald]

(x+1)^2 could be used tbh

[topaz pelican]

I'm dumb srey

[iron herald]

another sub method imo

[cold jolt]

np lol

o

[topaz pelican]

It shud be (x+1)^2

[cold jolt]

x^2 + 2x + 1 + 23
(x+1)^2 + 23

But its in a sqrt still

[empty raft]

u = x^2 + 2x + 24.

[iron herald]

so many sub methods lol

[cold jolt]

So we have x^2 + 2x -41 + 8sqrt(u)=0

[topaz pelican]

Yesh

[cold jolt]

Im at sqrt(u) = (x^2 +2x -41)/-8

If u is x^2 + 2x + 24 then sqrt(u) is the sqrt of that right? no +-

wait

(x^2 +2x -41)/-8 = sqrt(x^2 + 2x + 24)

[empty raft]

My work

Last one probably either i made a mistake or not actually a solution

Exterienious

However you spell it

@cold jolt

[topaz pelican]

Vro that was good one but make sure to not show entire solution sometimes coz it's against the rules ig

[cold jolt]

Ohhh

[topaz pelican]

Umm I found out another way

[cold jolt]

So i had to sub those x as u too

Yea ?

[cold jolt]

Tysm btw

[rough steeple]

@cold jolt has given 1 rep to @empty raft

[cold jolt]

Had my head hurting for an hour

[cold jolt]

Wait what

Thats an imaginary solution right

Cause have to sqrt -1

[cold jolt]

Yea before I was trying it on desmos but at that time I was squaring both sides t oget rid of radical

So it made two new solutions

Just checked and it matches yours

[topaz pelican]

So the equation might have no solutions idk ig I'll try

[empty raft]

Mb should’ve spoiled it or something

[topaz pelican]

Hmm nvm it's wrong

The above solution

Can't do like that ig

Coz u get no roots

[empty raft]

When doing this I forgot a minus at the very first step of u so i had to restart the entire thing

[warm escarp]

u = sqrt(x^2 + 2x + 24)
u^2 + 8u - 65 = 0
(u + 4)^2 - 81 = 0
u = -4 ± 9
u = -13 or u = 5

However, u = sqrt(x^2 + 2x + 24)

So u > 0, thus u = -13 can be rejected.

Now we only have u = 5

x^2 + 2x + 24 = 25
x^2 + 2x - 1 = 0
(x+1)^2 - 2 = 0

x = -1 ± sqrt(2)

[warm escarp]

@cold jolt Here’s my solution

The substitution I used is a little tricky to spot,

The thing that gave it away was how x^2 + 2x appeared in the radical and outside of the radical.

Hope this helps!

[cold jolt]

Thank you

[rough steeple]

@cold jolt has given 1 rep to @warm escarp

[cold jolt]

How do i close this

[warm escarp]

No problem!