Extraneous Solutions are my Weak Point

[NightShade]

Hi. I need someones help. I've got an assignment, and a few problems on it are the dreaded extraneous solutions. I know the basics of how to do the problems, but I continually get stuck. Can anyone possibly show me what I'm missing here? I will be indebted to you eternally. O.K. Here goes.

x= ^(24-10x)
Then you're supposed to square both sides of the problem.
x*2 = ^(24-10x)*2
Then my math book says to make a new equation.
x*2 = 24-10x

I'm so sorry, but on this problem, this is where I'm stuck at. My math book has an example problem, but at this point in that problem you subtract numbers. I can't think how you would subtract the 10x, and what other number would you subtract with it? Could you explain this to me please? Thank you!

It's the same with this problem too.

x = ^(11x-28)

x*2 = ^(11x-28)*2

x*2 = 11x-28

At this point, I get lost.

If you could possibly show me a simple way to get through these, please do so. You don't even have to use these problems, you can make up your own like it. I don't want you to have to do my work for me. I just want to figure out what I'm missing.

Thank you so very much!

Mon Stylo Est Le Baril D'un Pistolet; Rappelez-Moi De Quel Côté Vous Êtes

~Nuit D'Automne~

I'm sorry, I'm in a French mood today. I'll get over it later...

 

[Unco]

Hi. I need someones help. I've got an assignment, and a few problems on it are the dreaded extraneous solutions. I know the basics of how to do the problems, but I continually get stuck. Can anyone possibly show me what I'm missing here? I will be indebted to you eternally. O.K. Here goes.

x= ^(24-10x)
Then you're supposed to square both sides of the problem.
x*2 = ^(24-10x)*2
Then my math book says to make a new equation.
x*2 = 24-10x

Type the square root as sqrt(24 - 10x) and "x squared" as x^2.

\(\displaystyle \mbox{x^2 = 24 - 10x}\) is a quadratic, and we put it into the form
\(\displaystyle \mbox{ax^2 + bx + c = 0}\) because we can factorise (or apply the quadratic formula, etc) from there.

So subtract 24 and -10x from both sides to get:

\(\displaystyle \mbox{ x^2 + 10x - 24 = 0}\)

\(\displaystyle \mbox{ }\)NB: \(\displaystyle \mbox{ - (-10x) = +10x}\); and we could have subtracted \(\displaystyle \mbox{x^2}\) from both sides instead just as well.

Now solve for \(\displaystyle \mbox{x}\). (Incidentally, this quadratic can be factorised.)

Check to see if each solution (each value for x you obtain) works in the original equation (the one with the square root in it) by evaluating each side with x=... to see if the equation is true with that value for x, because squaring both sides can produce extraneous solution(s). There is only one value for \(\displaystyle \mbox{x}\) which satisfies the original equation.

 

[soroban]

Hello, NightShade!

\(\displaystyle x\:=\:\sqrt{24\,-\,10x}\)

Then you're supposed to square both sides of the problem.
\(\displaystyle \;\;\;x^2\:=\\sqrt{24\,-\,10x})^2\)

Then my math book says to make a new equation.
\(\displaystyle \;\;\;x^2\:=\:24\,-\,10x\)

I'm so sorry, but on this problem, this is where I'm stuck at.
My math book has an example problem, but at this point in that problem you subtract numbers.
I don't understand . . . "subtract numbers"?
I can't think how you would subtract the 10x, and what other number would you subtract with it?
I still don't understand . . .
Could you explain this to me please? Thank you!

We have: \(\displaystyle \:x^2\:=\:24\,-\,10x\)

Are you saying that you don't know how to get all the terms on one side??

\(\displaystyle \;\;\;x^2\,+\,10x\,-\,24\:=\:0\)

And that you don't know how to factor and solve a quadratic?

\(\displaystyle \;\;\;(x\,-\,2)(x\,+\,12)\:=\:0\;\;\Rightarrow\;\;x\,=\,2,\,\)-\(\displaystyle 12\)

 

[NightShade]

Thank You!

I still don't quite understand, but I figured out this problem. I'll try to figure it out though. I just need to take a closer look at the steps you guys showed. Thank you so much for helping me out. I really appreciate it.

Le temps est proche
Et le monde passe
Comme le soleil sourit vers le bas à vous
Et ma chanson est déjà chantée

~Nuit D'Automne~

 

[Denis]

Ombrage Du Soir, ecoute ton professeur avec attention :wink: