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# If k is an integer and x(x – k) = k + 1, what is the value of x?

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If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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01 Sep 2015, 21:51
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If k is an integer and x(x – k) = k + 1, what is the value of x?

(1) x < k
(2) x = 3 – k

Kudos for a correct solution.
[Reveal] Spoiler: OA

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Re: If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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01 Sep 2015, 23:43
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Bunuel wrote:
If k is an integer and x(x – k) = k + 1, what is the value of x?

(1) x < k
(2) x = 3 – k

Ans: C

Solution:
1) Option one does not give us any answer if we put the relation x<k into the given equation [Not sufficient]

2) putting x= 3-k we get a quardatic equation which look likes this
(3-k)(3-2k)=k+1
After solving it we get k=4,1

Now by putting these two values we get 2 different values of x for each value of k
When k= 4 x=5 , -1 and when k= 1 x= 2. ,-1. [Not Sufficient]

By combining both statements we get the value of x =-1
So both statement together are sufficient
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Re: If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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02 Sep 2015, 00:45
x(x – k) = k + 1 can be deduced to x^2-1 = k+kx -----> (x+1)(x-1)=k(x+1)-----> Finally k=x-1

St 1 : cannot help us evaluate x . Insuff
St 2 : substituting k we get x =2 . Suff

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Re: If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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02 Sep 2015, 02:29
Bunuel wrote:
If k is an integer and x(x – k) = k + 1, what is the value of x?

(1) x < k
(2) x = 3 – k

Kudos for a correct solution.

Statement one is insufficient as we get more than one value of x.

Statement 2 : substituting x = 3 - k in the equation given in the question. We get k = 1 and k = 4. However, on substitution in x = 3 - k we get two values of x 2 and -1. So insufficient.

When we combine both the statements the only value of x is 2.

So C
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If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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02 Sep 2015, 07:34
If k is an integer and x(x – k) = k + 1, what is the value of x?

(1) x < k
(2) x = 3 – k

1 ----> not sufficient there could be multiple values satisfying K>x

2 ----> x = 3-k
k = 3 - x putting this value in x(x – k) = k + 1, will get two values of x = 2,-1

not sufficient

1 and 2 ,,,,,, x =2 , k= 5 and x = -1 ,k =2
Still two value of x .
E
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If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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02 Sep 2015, 14:33
Solution:
Statement1: x<k .Not sufficient .

Statement2 : x = 3 - k. Substituting will give k=1,4. So, not sufficient.

Combined : k=1 is ruled out. So k=4 and x=-1.

Last edited by anudeep133 on 03 Sep 2015, 09:45, edited 1 time in total.

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If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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03 Sep 2015, 03:39
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and equations ensures a solution.

If k is an integer and x(x ? k) = k + 1, what is the value of x?

(1) x < k
(2) x = 3 ? k

In the original condition we have 2 variable (x,k) and 1 equation (x(x-k)=k+1) and in order to match the number of variables and equations we need 1 more equation. Since there is 1 each in 1) and 2), D is likely the answer.
In case of 1), x<k is not sufficient while in case of 2), substituting x=3-k gives us (3-k)(3-k-k)=k+1, (3-k)(3-2k)=k+1, 2k^2-9k+9=k+1, 2k^2-10k+8=0, k^2-5k+4=0, (k-4)(k-1)=0 k=1,4 therefore it is not unique. thus it is not sufficient. Using both 1) and 2) gives us unique answer k=4, x=-1, and therefore is sufficient.

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Last edited by MathRevolution on 06 Sep 2015, 19:48, edited 1 time in total.

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Re: If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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05 Sep 2015, 11:17
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Bunuel wrote:
If k is an integer and x(x – k) = k + 1, what is the value of x?

(1) x < k
(2) x = 3 – k

Kudos for a correct solution.

We know equal number of variables and equations ensures a solution.So I am gonna start to find the value of x from the equation of Statement (2) and x(x+k)=k+1 first Since inequalities of Statement (1)gives no solution

Statement (2) : x=3-k,or k=3-x,now put the value of k to x(x-k)=k+1 as below,

x(x-3+x)=3-x+1
or, 2x^2-3x=4-x
or, 2x^2-2x-4=0
or, x^2-x-2=0
or, x^2+x-2x-2=0
or,x(x+1)-2(x+1)=0
or, (x+1)(x-2)=0
So x=-1,or 2

Now, if x=-1 then k=3+1=4
and if x=2 then k=3-2=1

So Statement (2) is not sufficient

Now if we the the two statements when x=-1 and k=4 and x<k,the integration leads us to the Correct Answer C
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Re: If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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08 Sep 2015, 06:57
Hi Bunuel,

None of the above solutions show A as the answer, is the OA wrong ? If not, could you please explain the solution ?

Thanks.

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If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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08 Sep 2015, 07:14
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Swaroopdev wrote:
Hi Bunuel,

None of the above solutions show A as the answer, is the OA wrong ? If not, could you please explain the solution ?

Thanks.

I think everyone is missing out on a very important information.

Lets analyse the question:

Given equation: x(x-k) = k+1 ----> $$x^2-kx-k-1=0$$ --->$$x = \frac {k\pm \sqrt{k^2-4(-k-1)}}{2}$$

Thus you get 2 values of x as =$$\frac {k\pm(k+2)}{2}$$ ---> x = k+1 or x=-1

Per statement 1, x<k ---> x $$\neq$$k+1 ---> x = -1 is the only solution left. Thus this statement IS SUFFICIENT

Per statement 2, x = 3-k ---> substituting in the above given equation, you get, k = 4 or 1 ---> x = -1 or 2. Thus this statement is NOT sufficient.

This question is a classic "C-Trap" question.

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Re: If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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09 Sep 2015, 04:08
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AbdurRakib wrote:
We know equal number of variables and equations ensures a solution.

The statement is red is not 100% correct.

Consider the case:

a+2b = 10 and
2b+4b = 30

You have 2 equations and 2 equations and thus per your statement you should be able to solve for 'a' and 'b'. This is NOT true. You can not solve this system of equations as both the equations are essentially the same wrt the variables involved.

Instead the correct statement should read: We know equal number of variables and DISTINCT equations ensures a solution

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Re: If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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17 Jul 2016, 05:07
Hi!
I too picked C.

bunuel , chetan4u, empowergmat richi, magoosh mike, veritas..
plz help us on this
2) why answer c is wrong

thanks

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Re: If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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17 Jul 2016, 05:29
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Celestial09 wrote:
Hi!
I too picked C.

bunuel , chetan4u, empowergmat richi, magoosh mike, veritas..
plz help us on this
2) why answer c is wrong

thanks

I think the explanation as previously stated is the correct explanation for A.

x(x-k)=K+1
=> x^2-xk=k+1
=>
x^2-1=k(x+1)

or (x-1)(x+1)=k(x+1)

you cannot cancel x+1 on both sides directly.

Either we have k=(x-1) or x+1=0
=>
either x=k+1 or x=-1

Statement 1 says x<k => x <> k+1, thus x=-1, Hence it is sufficient.

Statement 2 says x=3-k => x+k=3. We cannot find out x here, as we don't know the value of K. Hence 2 is insufficient.

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Re: If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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17 Jul 2016, 06:28
1
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Celestial09 wrote:
Hi!
I too picked C.

bunuel , chetan4u, empowergmat richi, magoosh mike, veritas..
plz help us on this
2) why answer c is wrong

thanks

Did you check the solution at if-k-is-an-integer-and-x-x-k-k-1-what-is-the-value-of-x-204834.html#p1571106 ? Go through it and see if you still have any doubts.

When you do ask for help from an expert, make sure to mention what you have done on this question or what your thought process was.

Answer C is wrong as A is sufficient.

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If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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17 Jul 2016, 07:32
abhimahna wrote:
Celestial09 wrote:
Hi!
I too picked C.

bunuel , chetan4u, empowergmat richi, magoosh mike, veritas..
plz help us on this
2) why answer c is wrong

thanks

I think the explanation as previously stated is the correct explanation for A.

x(x-k)=K+1
=> x^2-xk=k+1
=>
x^2-1=k(x+1)

or (x-1)(x+1)=k(x+1)

you cannot cancel x+1 on both sides directly.

Either we have k=(x-1) or x+1=0
=>
either x=k+1 or x=-1

Statement 1 says x<k => x <> k+1, thus x=-1, Hence it is sufficient.

Statement 2 says x=3-k => x+k=3. We cannot find out x here, as we don't know the value of K. Hence 2 is insufficient.

-- Hit Kudos if you get the answer.

Why CANT I cancel
x+1 on both sides?
thanks

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Re: If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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17 Jul 2016, 07:39
Celestial09 wrote:
abhimahna wrote:
Celestial09 wrote:
Hi!
I too picked C.

bunuel , chetan4u, empowergmat richi, magoosh mike, veritas..
plz help us on this
2) why answer c is wrong

thanks

I think the explanation as previously stated is the correct explanation for A.

x(x-k)=K+1
=> x^2-xk=k+1
=>
x^2-1=k(x+1)

or (x-1)(x+1)=k(x+1)

you cannot cancel x+1 on both sides directly.

Either we have k=(x-1) or x+1=0
=>
either x=k+1 or x=-1

Statement 1 says x<k => x <> k+1, thus x=-1, Hence it is sufficient.

Statement 2 says x=3-k => x+k=3. We cannot find out x here, as we don't know the value of K. Hence 2 is insufficient.

-- Hit Kudos if you get the answer.

Why can I cancel
x+1 on both sides?
thanks

This is a trap that many of us get into. Isn't it possible that both sides of the equation are equal just because x was equal to -1, thus making 0 on both the sides.

Always remember, One should NEVER Cancel any common variables unless you are 100% sure about the actual value of that variable.

Let's say if we had 2 common on both sides, then we could have cancelled it as we know that it is NON ZERO Value. But we are not sure what the value of x is, Hence cancellation is NOT ALLOWED.

Hope that helps.

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If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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17 Jul 2016, 07:42
Expert's post
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Celestial09 wrote:
abhimahna wrote:
Celestial09 wrote:
Hi!
I too picked C.

bunuel , chetan4u, empowergmat richi, magoosh mike, veritas..
plz help us on this
2) why answer c is wrong

thanks

I think the explanation as previously stated is the correct explanation for A.

x(x-k)=K+1
=> x^2-xk=k+1
=>
x^2-1=k(x+1)

or (x-1)(x+1)=k(x+1)

you cannot cancel x+1 on both sides directly.

Either we have k=(x-1) or x+1=0
=>
either x=k+1 or x=-1

Statement 1 says x<k => x <> k+1, thus x=-1, Hence it is sufficient.

Statement 2 says x=3-k => x+k=3. We cannot find out x here, as we don't know the value of K. Hence 2 is insufficient.

-- Hit Kudos if you get the answer.

Why can I cancel
x+1 on both sides?
thanks

Because x + 1 can be 0 and we cannot divide by 0.

Never reduce equation by variable (or expression with variable), if you are not certain that variable (or expression with variable) doesn't equal to zero. We cannot divide by zero.

Never multiply (or reduce) inequality by variable (or expression with variable) if you don't know the sign of it or are not certain that variable (or expression with variable) doesn't equal to zero.

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If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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17 Jul 2016, 08:11
Celestial09 wrote:
abhimahna wrote:
Celestial09 wrote:
Hi!
I too picked C.

bunuel , chetan4u, empowergmat richi, magoosh mike, veritas..
plz help us on this
2) why answer c is wrong

thanks

I think the explanation as previously stated is the correct explanation for A.

x(x-k)=K+1
=> x^2-xk=k+1
=>
x^2-1=k(x+1)

or (x-1)(x+1)=k(x+1)

you cannot cancel x+1 on both sides directly.

Either we have k=(x-1) or x+1=0
=>
either x=k+1 or x=-1

Statement 1 says x x <> k+1, thus x=-1, Hence it is sufficient.

Statement 2 says x=3-k => x+k=3. We cannot find out x here, as we don't know the value of K. Hence 2 is insufficient.

-- Hit Kudos if you get the answer.

Why can I cancel
x+1 on both sides?
thanks

Another thing to note here is that the given expression in x is a quadratic one . Hence, you will have 2 solutions. By cancelling x+1 you are assuming that x = -1, thus reducing the number of solutions to 1 Instead of 2.

This is an incorrect assumption to make in DS questions as the stem nowhere mentions whether x= -1 is an acceptable solution or not.

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Re: If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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17 Jul 2016, 23:57
A

1)only x=-1 and k=0 suffice.try all values on +v e and -ve side only k=0 and x=-1 suffice .

2)2 values of k

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Re: If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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18 Jul 2016, 05:22
hsbinfy wrote:
A

1)only x=-1 and k=0 suffice.try all values on +v e and -ve side only k=0 and x=-1 suffice .

2)2 values of k

Statement in red above is not complete. x=1 with k=0 also works in the given expression. The only good way to look at is to factorize the given expression and see which of the 2 possible solutions makes most sense.

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Re: If k is an integer and x(x – k) = k + 1, what is the value of x?   [#permalink] 18 Jul 2016, 05:22

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