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# Does the equation y = (x – p)(x – q) intercept the x-axis at

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Does the equation y = (x – p)(x – q) intercept the x-axis at [#permalink]

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09 Jun 2012, 16:24
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Does the equation y = (x – p)(x – q) intercept the x-axis at the point (2,0)?

(1) pq = -8

(2) -2 – p = q
[Reveal] Spoiler: OA

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09 Jun 2012, 17:19
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y=(x-p)(x-q)

Basically, when y=0 and x=2, does equation balance?

0 = (2-p)(2-q)
0 = 4-2q-2p+qp

In order to know this, we need to know q & p

1) pq = -8
This only tell us p or q, but not both.

BCE

2) -2 - p = q
This only tell us p or q, but not both.

1+2) We can determine both p & q through these two independent equations.

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Re: Does the equation y = (x – p)(x – q) intercept the x-axis at [#permalink]

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10 Jun 2012, 01:13
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Does the equation y = (x – p)(x – q) intercept the x-axis at the point (2,0)?

x-intercepts of the graph $$y=(x-p)(x-q)$$ is the values of $$x$$ for which $$(x-p)(x-q)=0$$. So, the x-intercepts are $$(p, 0)$$ and $$(q, 0)$$. The question basically asks whether either $$p$$ or $$q$$ equals 2.

(1) pq = -8. Not sufficient to say whether p or q equals 2.

(2) -2 – p = q. Not sufficient to say whether p or q equals 2.

(1)+(2) Solving $$pq=-8$$ and $$-2-p=q$$ gives us that either $$p=-4$$ and $$q=2$$ OR $$p=2$$ and $$q=-4$$. In ether case one of the unknowns is 2, so $$y=(x-p)(x-q)$$ intercepts the x-axis at the point (2,0). Sufficient.

Hope it's clear.
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Re: Does the equation y = (x – p)(x – q) intercept the x-axis at [#permalink]

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10 Jun 2012, 01:15
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Bunuel wrote:
Does the equation y = (x – p)(x – q) intercept the x-axis at the point (2,0)?

x-intercepts of the graph $$y=(x-p)(x-q)$$ is the values of $$x$$ for which $$(x-p)(x-q)=0$$. So, the x-intercepts are $$(p, 0)$$ and $$(q, 0)$$. The question basically asks whether either $$p$$ or $$q$$ equals 2.

(1) pq = -8. Not sufficient to say whether p or q equals 2.

(2) -2 – p = q. Not sufficient to say whether p or q equals 2.

(1)+(2) Solving $$pq=-8$$ and $$-2-p=q$$ gives us that either $$p=-4$$ and $$q=2$$ OR $$p=2$$ and $$q=-4$$. In ether case one of the unknowns is 2, so $$y=(x-p)(x-q)$$ intercepts the x-axis at the point (2,0). Sufficient.

Hope it's clear.

Damn, you're good! Was my approach right at all? Sometimes I wish a had an identical twin who could just get a 50 on my Quant section for me while I do the Verbal section, haha.
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Does the equation y = (x – p)(x – q) intercept the x-axis at the [#permalink]

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13 Jan 2013, 11:30
C.

If y = 0,

this reduces to a quadratic equation.
sum of roots = 2, product of roots = -8.
Thus the roots are 4 and -2.
Line passes through (4,0) and (-2,0)
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Re: Does the equation y = (x – p)(x – q) intercept the x-axis at [#permalink]

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Re: Does the equation y = (x – p)(x – q) intercept the x-axis at [#permalink]

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30 May 2015, 05:51
You can also distribute the given equation straight away:
y = x^2 - xq - px + pq
This can be factored to:
y = x^2 - x(p+q) + pq
From this, it's easy to recognise that if we know 'p + q' AND 'pq' then we know the line's equation, and can figure out the answer. No further calculation is necessary. As is often the case with Manhattan GMAT questions, rearranging the question is heavily rewarded.
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Why is Statement (1) NOT sufficient? "Does the equation y = ( x – p)( [#permalink]

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31 May 2015, 18:18
Does the equation y = ( x – p)( x – q) intercept the x-axis at the point (2,0)?

(1) pq = -8

(2) -2 – p = q

[Reveal] Spoiler:
C
but:

My analysis tells me that (1) should be sufficient based on these steps:

Plug in (2,0) in the equation and expand:

0 = (2 - p)(2 - q)
0 = 2^2 - 2q - 2p + pq

Now, sub in (1) pq = -8

0 = 4 - 2q - 2p -8
4 = -2q - 2p
-2 = q + p
Hence: -2 - p = q --> same as statement (2)... so if I'm able to derive statement (2) just using information in (1), can't I plug "-2 = q + p" into pq = -8 and solve? Isn't this the same as using both statements except you can only use the first one to get the info in the second, thus making (1) sufficient?

What am I missing here?

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Re: Why is Statement (1) NOT sufficient? "Does the equation y = ( x – p)( [#permalink]

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31 May 2015, 22:24
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Great question. The problem is that you are presupposing the truth of what you are trying to figure out. The fact that statement 2 equals what you get when you manipulate statement 1 means that together they prove that the equation does intercept the x-axis at this point. Think about what you did, but using the random point (4,0). That would give you:

0 = (4-p)(4-q)
0 = 4^2 - 4q - 4p + pq
Sub in (1)
0 = 16 - 4q - 4p - 8
-8 = -4q - 4p
2 = q + p
2-p = q

This now leaves you with a different equation, when we assume that the equation intercepts at point (4,0). Therefore the fact that this equation equals what is given in statement 2 when you plug in (2,0) indicates that the equation intercepts at the point (2,0), and thus you need both statements in order to know that.

I hope this helps!
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Re: Why is Statement (1) NOT sufficient? "Does the equation y = ( x – p)( [#permalink]

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31 May 2015, 22:32
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Hello,

Statement(2) is derived while substituting answer in Statement (1). So we are assuming y=(x-p) (x-q) at (2,0) to be true.It will only be possible if and only if Statement (2) is true as you have already derived. Hence we need both .
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Re: Why is Statement (1) NOT sufficient? "Does the equation y = ( x – p)( [#permalink]

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31 May 2015, 22:47
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Hi torontoclub15,

When you say that statement-I is sufficient to answer the question, you mean that information given in the question statement along with the information given in st-I is adequate to give you a unique answer.

When you are analyzing any one of the statements, pretend the other statement to be non-existent to avoid carrying over of the information. Any dependency on the other statement is never going to give you the answer as A or B.

In this question, since you are putting (2,0) in the equation ,you are assuming that the equation intercepts the x-axis at (2,0). This assumption and using the information given in st-I results in the equation -2 -p = q. For the equation to intercept the x-axis at (2,0), the above equation of -2 - p = q should be true. But you don't know the values of p and q or p + q and hence can't say for sure if the equation is true or not. Hence you can't say if the equation intercepts the x-axis at (2,0).

Similarly for st-II, if you assume that the equation passes through (2,0) and use the information given in st-II you would end up at pq= -8. Again using st-II alone you can't say for sure if pq = -8?. Hence st-II also is not sufficient to answer the question.

You would observe that both the statements need each other to give you a definite answer if the equation passes through (2,0). Hence the answer has to be C.

For avoiding such errors we recommend the following 5-step process to solve a DS question

Step-I: List down the given info

Step-II: Analyze the given info

Step-III: Analyze statement-I independently

Step-IV: Analyze statement-II independently

Step-V Combine both statements if needed

Following these steps have the following advantages:

a. You analyze the question statement to narrow down to the exact information you are looking in the statements. This helps in avoiding unnecessary analysis.

b. By analyzing statements independently you make sure that there is no carrying over of the information from one statement to the other before you reach step-V

Hope this helps

Regards
Harsh
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Re: Why is Statement (1) NOT sufficient? "Does the equation y = ( x – p)( [#permalink]

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31 May 2015, 22:49
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torontoclub15 wrote:
Does the equation y = ( x – p)( x – q) intercept the x-axis at the point (2,0)?

(1) pq = -8

(2) -2 – p = q

[Reveal] Spoiler:
C
but:

My analysis tells me that (1) should be sufficient based on these steps:

Plug in (2,0) in the equation and expand:

0 = (2 - p)(2 - q)
0 = 2^2 - 2q - 2p + pq

Now, sub in (1) pq = -8

0 = 4 - 2q - 2p -8
4 = -2q - 2p
-2 = q + p
Hence: -2 - p = q --> same as statement (2)... so if I'm able to derive statement (2) just using information in (1), can't I plug "-2 = q + p" into pq = -8 and solve? Isn't this the same as using both statements except you can only use the first one to get the info in the second, thus making (1) sufficient?

What am I missing here?

Hi,

For Data Sufficiency questions, you are not required to find answers. You just have to tell whether the statements given are sufficient to

At first you have to treat each statements independently. You should not take any information from statement 2 when you are trying to
figure out if statement 1 alone is sufficient and vice versa. (Basically you have to assume the other statement is not there when you are solving
for one statement alone.)
If each statement alone does not give you answer then you have to combine both and check.

A. Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.
B. Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question
C. Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient
D. Each statement alone is sufficient
E. Statements 1 and 2 together are not sufficient.

What is the question here?
Does the equation y = ( x – p)( x – q) intercept the x-axis at the point (2,0)? You are required to tell if the statements given are sufficient to answer the questions.
i.e. y = x^2 - x(p+q) + pq . You need information about both 'p+q' and 'pq' to solve for this.

Take statement 1 alone. Are you able to answer the above question just by taking statement 1 alone.?
No. Because you know pq = -8.But you don't know what is p+q. When you are checking for statement 1 alone you should not take any info from statement 2.

Take statement 2 alone. Are you able to answer the above question just by taking statement 2 alone.?
No.Because you know p+ q = -2 But you don't know what is pq. When you are checking for statement 2 alone you should not take any info from statement 1.

When each statement alone is not giving you any answer. You have to combine both.
Now you will see that there is information about p+q AND pq which is sufficient to answer the question.

Hope you understood.
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Re: Does the equation y = (x – p)(x – q) intercept the x-axis at [#permalink]

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01 Jun 2015, 08:02
Appreciate your replies. It's very clear now. Thanks so much!
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Does the equation y = (x – p)(x – q) intercept the x-axis at [#permalink]

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01 Dec 2015, 01:26
alchemist009 wrote:
Does the equation y = (x – p)(x – q) intercept the x-axis at the point (2,0)?

(1) pq = -8

(2) -2 – p = q

Question: $$(x – p)(x – q) = x^2-x(p+q)+pq and x=2 and y=0 --> 4-2(p+q)+pq=0 --> 2(p+q)=4+pq ?$$

(1) pq=-8 -> 2(p+q)=-4, p+q=-2 Not sufficient

(2) p+q=-2 -> 4-2*(-2)+pq=0, pq=-8 Not sufficient

(1)+(2) 4-2*(-2)+(-8)=0 Yes

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Re: Does the equation y = (x – p)(x – q) intercept the x-axis at [#permalink]

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01 Dec 2015, 05:52
Bunuel wrote:
Does the equation y = (x – p)(x – q) intercept the x-axis at the point (2,0)?

x-intercepts of the graph $$y=(x-p)(x-q)$$ is the values of $$x$$ for which $$(x-p)(x-q)=0$$. So, the x-intercepts are $$(p, 0)$$ and $$(q, 0)$$. The question basically asks whether either $$p$$ or $$q$$ equals 2.

(1) pq = -8. Not sufficient to say whether p or q equals 2.

(2) -2 – p = q. Not sufficient to say whether p or q equals 2.

(1)+(2) Solving $$pq=-8$$ and $$-2-p=q$$ gives us that either $$p=-4$$ and $$q=2$$ OR $$p=2$$ and $$q=-4$$. In ether case one of the unknowns is 2, so $$y=(x-p)(x-q)$$ intercepts the x-axis at the point (2,0). Sufficient.

Hope it's clear.

Hi Bunuel

When i do a quad equation,

i get (x+4)(x-2)=0

this means that either x=2 or -4

how can we conclude it to be sufficient, as it could be either 2,0 or -4,0

thanks
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Re: Does the equation y = (x – p)(x – q) intercept the x-axis at [#permalink]

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01 Dec 2015, 06:45
rahulkashyap wrote:
Bunuel wrote:
Does the equation y = (x – p)(x – q) intercept the x-axis at the point (2,0)?

x-intercepts of the graph $$y=(x-p)(x-q)$$ is the values of $$x$$ for which $$(x-p)(x-q)=0$$. So, the x-intercepts are $$(p, 0)$$ and $$(q, 0)$$. The question basically asks whether either $$p$$ or $$q$$ equals 2.

(1) pq = -8. Not sufficient to say whether p or q equals 2.

(2) -2 – p = q. Not sufficient to say whether p or q equals 2.

(1)+(2) Solving $$pq=-8$$ and $$-2-p=q$$ gives us that either $$p=-4$$ and $$q=2$$ OR $$p=2$$ and $$q=-4$$. In ether case one of the unknowns is 2, so $$y=(x-p)(x-q)$$ intercepts the x-axis at the point (2,0). Sufficient.

Hope it's clear.

Hi Bunuel

When i do a quad equation,

i get (x+4)(x-2)=0

this means that either x=2 or -4

how can we conclude it to be sufficient, as it could be either 2,0 or -4,0

thanks

The question asks whether either $$p$$ or $$q$$ equals 2. Solving gives two solution sets:

1. p=-4 and q=2.
Or:
2. p=2 and q=−4.

So, in the first case q=2 and in the second case p=2.
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Re: Does the equation y = (x – p)(x – q) intercept the x-axis at [#permalink]

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01 Dec 2015, 07:11
Bunuel

But the solution is for "x"
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Re: Does the equation y = (x – p)(x – q) intercept the x-axis at [#permalink]

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01 Dec 2015, 07:15
rahulkashyap wrote:
Bunuel

But the solution is for "x"

I don't understand what you mean by this. Can you please re-read the solutions given above?
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Re: Does the equation y = (x – p)(x – q) intercept the x-axis at [#permalink]

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01 Dec 2015, 08:06
Bunuel wrote:
rahulkashyap wrote:
Bunuel

But the solution is for "x"

I don't understand what you mean by this. Can you please re-read the solutions given above?

Well, the equation is : y=(x−p)(x−q)

to find the x intercept, I plugged in y=0
this gives me the equation x^2 - qx - px +pq= 0

this, on using (1) and (2)

i get : x^2 + 2x - 8 =0

we get : (x+4)(x-2)= 0

so, x=2 or -4 from solving the quad equation
therefore, (2,0) or (-4,0)
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Re: Does the equation y = (x – p)(x – q) intercept the x-axis at [#permalink]

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03 Dec 2015, 07:38
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 independent equations ensures a solution.

Does the equation y = (x – p)(x – q) intercept the x-axis at the point (2,0)?

(1) pq = -8

(2) -2 – p = q

If we modify the qusetion, 0=(2-p)(2-q)? and 0=4-2p-2q+pq?. There are 2 variables (p,q) and 2 equations are given by the 2 conditions, so there is high chance (C) will be the answer.
Looking at the conditions together,
0=4-2(p+q)+pq? --> 0=4-2(-2)-8? --> 0=4+4-8=0? This answers the question 'yes' and is therefore sufficient and the answer becomes (C).

For cases where we need 2 more equations, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E.
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Re: Does the equation y = (x – p)(x – q) intercept the x-axis at   [#permalink] 03 Dec 2015, 07:38

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