If (x - 1)^2 = 400 which of the following could be the value

Hi,

this is an interesting problem regarding the GMAT Club Math Book. I studied it and on page 15 it says:

When the GMAT provides the square root sign for an even root, the only accepted answer is the positive root. That is, NOT +5 or -5. In contrast, the equation x^2 has TWO solutions, +5 and -5. Even roots have only a positive value on the GMAT.

Due to this fact (which of course does not include this particular situation, because it's talking about the "sign"), I didn’t consider the negative root.
Maybe a phrase, which could be improved on the next issue .

That statement from the book still holds good, it says "when GMAT provides the square root sign" whereas in this problem you were explicitly calculating the square root of 400 rather than GMAT providing \(\sqrt{400}\).
Hope that helps.

Hi there, where did the -4 come from? i.e.
\(x-5=20-4=16\) or \(x-5=-20-4=-24\)

Thanks.

hi, remember we have calculated the value of x-1=20 or -20. ------------1)
x-5 = (x-1)-4 now from 1, put the value of x-1 as 20 or -20
= 20-4 =16
or x-5 = (x-1)-4 = -20-4 = -24

i hope it helps.

Hi there, where did the -4 come from? i.e.
\(x-5=20-4=16\) or \(x-5=-20-4=-24\)

Thanks.[/quote]

hi, remember we have calculated the value of x-1=20 or -20. ------------1)
x-5 = (x-1)-4 now from 1, put the value of x-1 as 20 or -20
= 20-4 =16
or x-5 = (x-1)-4 = -20-4 = -24

i hope it helps.[/quote]

Yep great - understand now. Appreciate your help!

Could we also solve this like this:

(x-1)^2 = (x-1)(x-1)?

I tried like this at first, but didn't end up well.. Unless I miscalculated sth somewhere.

Thank you.

Hi All,

GMAT writers often provide little 'hints' in the wording of the question that can help you to avoid some of the work.

Here, notice the phrase "...which of the following COULD be the value...." That's an interesting way to phrase a question - it's NOT asking "...what IS the value...." - it's asking "what COULD be the value...."

This implies that there's MORE than 1 answer AND that the 'obvious' answer is not the one that's going to be listed.

With (X-1)^2 = 400

I know there are two solutions (because of the 'squared sign')...

(X-1) COULD = 20 or -20

From the wording of the prompt though, it's likely that the "-20 option" is the one that we supposed to be going after, since that's the less obvious solution. Obviously, there's nothing wrong with being thorough and finding BOTH answers, but the question didn't ASK for that, so you have to be mindful about how much extra work you might be doing (and how much extra time you might be spending) on a given question, especially if you have a pacing problem.

GMAT assassins aren't born, they're made,
Rich

Hi pacifist85,

While you could convert the equation into a Quadratic, I don't think that you'll find that method to be particularly fast nor easy....

(X-1)^2 = 400
(X-1)(X-1) = 400

X^2 - 2X + 1 = 400

X^2 - 2X - 399 = 0
(X - 21)(X + 19) = 0

X = 21 or -19

Now, contrast the above with the following....

(something)^2 = 400
(something) = +20 or -20

(X - 1) = 20
or
(X - 1) = -20

X = 21 or -19

GMAT assassins aren't born, they're made,
Rich

\((x - 1)^2 = 400\)

\(x - 1 = \pm 20\)

\(x = 1 \pm 20 = 21 OR -19\)

x - 5 = 21-5 = 16 OR -19-5 = -24

Answer = C

Here is an another approach

20^2 =400
thus (x-1)^2 = 20^2
(x-1)^2 - 20^2 = 0
now we know that a^2-b^2 = (a-b)(a+b)
here x-1 =a and 20=b

thus we have
(x-1-20)(x-1+20)=0
(x-21)(x+19)=0
x=21 or x=-19

thus x-5=21-5 =16 or -19-5 =-24

Taking the root of both sides is definitely a better approach. Finding the roots of the quadratics can drive one insane!

\({ \left( x-1 \right) }^{ 2 }=400\\ \sqrt { { \left( x-1 \right) }^{ 2 } } =\sqrt { 400 } \\ \left| x-1 \right| =20\\ +\left( x-1 \right) =20\quad \leftrightarrow \quad -\left( x-1 \right) =20\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad or\\ { \left( x-1 \right) }^{ 2 }=400\\ \left( x-1 \right) \left( x-1 \right) =400\)

\((x - 1 ) =\) + \(20\)

If\(x - 1 = +20\) ; \(x = 21 ; So, x - 5 = 16\)

If \(x - 1 = -20\) ; \(x = -19; So, x - 5 = -24\)

Thus, answer must be (C) -24

(x-1)^2= 400
(x-1)^2= (20)^2

(x-1)^2 -(20)^2= 0

Use identity a^2-b^2= (a+b)*(a-b)

(x-1+20)(x-1-20)=0
(x+19) (x-21)=0

Equate the factorized form to zero.

x= -19 or x= 21

Plug in (x-5) and check the options.


If I took (-20)^2

(x-1)^2 - (-20)^2= 0

(x-1-20)* (x-1+20)= 0

(x-21) (x+19)= 0

x= 21 or x= -19


Same answer.

This identity: a^2-b^2= (a+b)(a-b) comes in handy in many questions.

I think there is very simple solution to this question no need to do any algebra

we can see that (x - 1)^2 = 400, the first thought occured to me here (x-1)^2 x = should be 21, but from answer choices you can`t see 21-5 = 16
so the only option is x = - 19 (-19-1)^2 which still yields -20 just negative in this case. but negative sign dissapears when raised to even power.

hence -19-5 = -24

so B

Given: (x - 1)^2 = 400

Asked: Which of the following could be the value of x - 5?

x-1 = +-20

x-5 = +-20 -4

x- 5 = 20-4 or -20-4 => 16 or -24

IMO C

Correct option : C

\((x-1)^2\) = 400
\(x^2 - 2x - 1\) = 400
\(x^2\) - 21x + 19x - 399 = 0
(x-21)(x+19) = 0
x = 21 or x =(-19)

x-5 = 21-5 = 16
x-5 = -19-5 = -24

Method 1:
(x - 1)^2 = 400
(x - 1) = +/- 20
(x - 5)
= ([x - 1] - 4) = +/- 20
(-20 - 4) = -24 or (20 - 4) = 16

Method 2:
(x - 1)^2 = 400
(x^2 - 2x - 1) = 400
(x^2 - 2x - 399) = 0
(x^2 - 21x + 19x - 399) = 0
(x - 21) (x +19) = 0
x = 21 or x = -19
x - 5 = 21 - 5 = 16
or
x - 5 = -16 - 5 = -24