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

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.

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)
Hence the answer is NO

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.

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

(1) pq = -8

(2) -2 – p = q

The answer is 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?

Kudos for any help please!

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!

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 .

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

Hi,

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

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.
Please look at the answer choices:

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.

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

Answer C

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.

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
on solving this quad eq,

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

so, x=2 or -4 from solving the quad equation
therefore, (2,0) or (-4,0)

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.

Solving the given equation takes from of a quadratic equation i.e the equation of a parabola. A parabola meets the x axis and in order to find so we need the entire quadratic equation.

The equation looks as this X^2 - (P+Q)X + PQ

We get these two components from Statements 1 and 2. Hence, C

Also, this equation is in the form of sum of roots and product of roots.

Statement 1

pq= -8

P and Q could take on multiple values- and these values would affect whether x intercepts the x axis at point (2,0)

4, -2
8, -1
-8, 1 etc

Statement 2

There are two variables; no ratio is given, no criteria is given- cannot solve

insuff

Statement 1 & 2

With both statements we can calculate the exact values of p and q

Thus
"C"

Another way to solve :

We first have to put the (2,0) in the given equation.

So we just have to DETERMINE whether 0=(2-p)(2-q). Because if this is true then the equation intercepts the x-axis at (2,0).

(1) pq = -8

This gives us 4 cases :

p= 2, q= -4
p= -4, q= 2
p= -2, q = 4,
p= 4, q= -2

Put these values in the equation and you will find that sometimes the value is equal to 0 and sometimes NOT.

Therefore, Insufficient.


(2) -2-p=q

From this, q can be anything. We need a specific case.

Hence, Insufficient.

Consider (1) + (2), Combine the two equations and SOLVE for p and q :

p(-2-p)=-8
---> -2p -p^2 = -8
----> p^2 + 2p -8 = 0 (SHIFTED ALL ON ONE SIDE )
-----> p^2 -2p + 4p - 8 = 0
-------> p(p-2) + 4(p-2) = 0
Therefore, (p+4) (p-2) = 0
It means that p = -4 OR p=2

If p = -4, q = 2 [AS STATED IN (1) ] ------(A)
If p=2, q=-4 [AS STATED IN (1) ] ------ (B)

When you put (A) AND (B) in the MAIN EQUATION, Then IT SATISFIES the equation meaning that :

Putting (A),
0 = [2-(-4)] x (2-2)
0 = 6 x 0
0 = 0 ----> TRUE

Putting (B),

0 = (2-2) X [ 2-(-4) ]
0 = 0 x 6
0 = 0 ------> AGAIN TRUE

THEREFORE, y=(x-p)(x-q) Intercepts the x-axis at point (2,0).

ANSWER : (C)

Bunuel Does this approach look good?

Equation y = x^2 - (p+q)x + pq intercept the x-axis at the point (2,0) if
0 = 4 - 2(p+q) + pq
=> 2(p+q) = pq +4
We need to prove 2(p+q) = pq+4

Statement 1 provides pq = -8
2(p+q) = -8+4
(p+q) = -2
p & q can take multiple values. INSUFFICIENT.

Statement 2 provides -2 -p = q
p+q = -2
-2*2 = pq +4
pq = -8
p & q can take multiple values. INSUFFICIENT.

Statement 1 & Statement 2 combined give
pq = -8 and p+q = -2
-4 = -8 + 4
Statement 1 & Statement 2 together are SUFFICIENT.

IMO C