Algebra: Solving Equations by Factoring

Question

Hi all, I'm going through The Official Guide for GMAT Review (12th Edition). Unfortunately for me, I didn't take any math courses beyond what was required to graduate from high school.

I've hit a wall with Algebra: Solving Equations by Factoring. I'm given the following example:

x^3 – 2x^2 + x = –5(x – 1)^2:

x^3 - 2x^2 + x + 5(x – 1)^2 = 0 
x(x^2 + 2x + 1) + 5(x – 1)^2 = 0 
x(x – 1)^2 + 5(x – 1)^2 = 0 
(x + 5)(x – 1)^2 = 0 
x + 5 = 0 or (x – 1)^2 = 0 
x = –5 or x = 1

I understand the first step, moving all of the expressions to one side, with 0 on the other side. I understand the next step, factoring the X. After that, I'm clueless. How the heck do you get from x(x^2 + 2x + 1) + 5(x – 1)^2 = 0 to x(x – 1)^2 + 5(x – 1)^2 = 0 ?

I tried to work backwards and did (x-1)^2. This came out to x^2  -  2x + 1... NOT the (x^2  +  2x +1) in the previous step. What am I doing wrong? Why and how are they doing what they are doing? I'm also completely clueless about the next step.

If someone can please hold my hand and explain this to me (keeping in mind that I'm a complete math moron and roll my eyes whenever the books say "so we can deduce that..."), it would be much appreciated!

KarishmaB · Answer

x^3- 2x^2 + x = - 5(x - 1)^2 :

x^3 - 2x^2 + x + 5(x - 1)^2 = 0

Here, you take x common from the first 3 terms. You get:

x(x^2  -  2x + 1) + 5(x - 1)^2 = 0

Now,  x^2 - 2x + 1 = (x - 1)^2  This is an algebraic identity which you should learn. Anyway you can easily deduce it as you already did when you got (x - 1)^2 = x^2 - 2x + 1 as you mentioned below

x(x - 1)^2 + 5(x - 1)^2 = 0

Here, (x - 1)^2 is taken common so you are left with x from the first term and 5 from the second one.

(x + 5)(x - 1)^2 = 0

Product of 2 factors is 0. It means at least one of them has to be 0. e.g. you say if x*y = 0, either x or y or both should be 0. Otherwise you will not get the product 0. Same logic with factors.

x + 5 = 0 or (x – 1)^2 = 0 
x = –5 or x = 1

Piece of Advice: OG12 will not give you the required theory. It is good for practice problems only. Since your concepts are rusty, get some basic algebra book e.g. we explain basic Math fundamentals in our book 'Math Essentials'. Check that out or any other basic Math book first.

GSDster · Answer

VeritasPrepKarishma, thank you very much!

Yes, I started to realize that doing the OG was probably not the best idea, especially since I need a lot of help with the theories. WIth that said, a friend loaned me the Manhattan GMAT books. I've been going through the number properties section, and that algebraic identity came up in a problem, but they had never explained it anywhere in the book. Very frustrating. And hard to research since I don't know what it's called... I didn't even know it was an "algebraic identity" until you mentioned it.

Is there anywhere where I can find other similar algebraic identities? While it's possible to deduce them, it's taking me forever to figure out problems because I just don't know about them. For example, I ran into...

"If n is a positive integer, is n^3 - n divisible by 4 (data sufficiency question)?"

I tried to simplify this by turning it into...
n(n^2 - n)

From there, I was stumped. Only after looking at the answer, I found out that I could and should have turned n(n^2 - n) into n(n - 1)(n + 1), which I could rearrange to be (n - 1)n(n + 1)... bingo! They're consecutive numbers! Now it makes sense. But as far as that step of turning the original equation into n(n - 1)(n + 1), I'm not clear on the rules and conditions that allow me to do this.

If there's somewhere where I can find a list and explanation of such algebraic identities/tricks/whatever they're called, please let me know!

KarishmaB · Answer

The identities and other algebra basics have been discussed in the Veritas Algebra book though if you are just looking for the identities you need to know, here they are:

1. (x + y)^2 = x^2 + y^2 + 2xy 
 2. (x - y)^2 = x^2 + y^2 - 2xy 
 3. x^2 - y^2 = (x + y)(x - y)  (called difference of squares - most important one. You should be able to recognize that  x^2 - 9 = (x + 3)(x - 3)  etc)

and if you would like to know how they are obtained, 
 (x + y)^2 = (x+y)(x+y) = x^2 + xy + yx + y^2 = x^2 + y^2 + 2xy

(x - y)^2 = (x-y)(x-y) = x^2 - xy - yx + y^2 = x^2 + y^2 - 2xy

(x+y)(x-y) = x^2 - xy + yx - y^2 = x^2 - y^2

GSDster · Answer

Thanks Karishma, you've been a huge help!

Regarding my original problem, is it safe to say that it was an error in the OG? Your explanation seems to confirm that x(x^2 + 2x + 1) + 5(x – 1)^2 = 0 really should have been x(x^2 - 2x + 1) + 5(x – 1)^2 = 0.

Is there an errata list somewhere?

EDIT: Not sure why, but whenever I try to use the math symbols tag, it turns my subtraction signs into a mess.

KarishmaB · Answer

x^3 – 2x^2 + x = –5(x – 1)^2:

x^3 - 2x^2  + x + 5(x – 1)^2 = 0

Till this step, you have a '-' sign which is fine. I don't know how you got a '+' in the next step. When you take x common, the '-' sign stays as it is... Perhaps a typo at your end or in OG...

x( x^2 + 2x  + 1) + 5(x – 1)^2 = 0

It should be x( x^2 - 2x  + 1) + 5(x – 1)^2 = 0
It is fine after this.

x(x – 1)^2 + 5(x – 1)^2 = 0 
(x + 5)(x – 1)^2 = 0 
x + 5 = 0 or (x – 1)^2 = 0

P.S. - Use a hyphen for '-' sign. It will stay put.

GSDster · Answer

Yes, I double-checked the OG and they went from x^3 - 2x^2 + x + 5(x-1)^2 = 0 to x(x^2 + 2x + 1)+4(x-1)^2=0 . So, I guess it was a typo on their end. So frustrating. :roll:

I sort of figured out the problem I was having with the math tags... apparently, copy and pasting equations into here can mess things up. I retyped the equations manually and they worked fine. Thanks again for all your help!

Mackam1234 · Answer

karishma,

I am having problems with this as well. I don't understand when you say

"Here (x-1)^2 is taken common so you are left with x from the first term and 5 from the second one. I still don't understand what happened with the first (x-1)^2. I see where the x+5 came from though.  VeritasPrepKarishma

Also I seem to be a bit rusty on my math theory. Can you suggest a good book to look at? I haven't really done any math in the last 10 years, since I left school.

ENGRTOMBA2018 · Answer

Try Foundations of GMAT Math by Manhattan Prep, it is a good book dealing with basics to tackle other more advanced concepts.

After you get:

x(x-1)^2+5(x-1)^2=0

Take, (x-1)^2 common from both the terms (by terms I mean,  x(x-1)^2  is 1 term and  5(x-1)^2  is the 2nd term).

So when we take (x-1)^2 common from the 2 terms, we will x from 1st term and 5 from the 2nd term left as shown below:

(x-1)^2 (x+5) = 0  (you can check this by applying the 'formula'  a(b+c) = a.b + a.c  , where  a = (x-1)^2  ,  b = x  and  c = 5 )

Finally, as you get something similar to a.b = 0 ---> this means that 1 of the 3 cases are possible: either a = 0 or b =0 or a =b=0 (dont worry about this case for now!))

Thus, either a = 0 --->  (x-1)^2 = 0  --->  x-1 = 0  --->  x =1  or

b = 0 --->  (x+5) = 0 --->  x = -5

Hope this helps.

KarishmaB · Answer

Conceptually, it is the same as saying

xy + xz = x(y + z)  ---- you take x common out of the two terms

Similarly, if you have 
 (x+1)y + (x+1)z = (x+1)(y + z)  ---- again, you take (x+1) common

Here, you have:

x(x^2 - 2x + 1) + 5(x – 1)^2 = 0

Now,  (x^2 - 2x + 1) = (x-1)^2  (because  (a - b)^2 = a^2 - 2ab + b^2 )

x(x - 1)^2 + 5(x - 1)^2 = 0

Take  (x-1)^2  common

(x - 1)^2*(x + 5) = 0

I would suggest you to check out your high school book before you work on any GMAT curriculum. Once you revise the basics of quadratics, linear equations etc, GMAT discussions will make much more sense.

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

Algebra: Solving Equations by Factoring

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards