DS problem

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prernamalhotra: Senior | Next Rank: 100 Posts; Posts: 44; Joined: Wed Apr 02, 2014 2:01 amFollowed by: 1 member; Thanked: 2 times

by prernamalhotra » Mon Jul 14, 2014 4:04 am

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

(1) pq = -8

(2) -2 - p = q

Thank you,
Prerna

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Source: Beat The GMAT — Data Sufficiency | Mon Jul 14, 2014 4:04 am

ianurag: Newbie | Next Rank: 10 Posts; Posts: 5; Joined: Sat Sep 14, 2013 11:33 pm

by ianurag » Mon Jul 14, 2014 5:13 am

C

Both equation required to solve

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theCodeToGMAT: Legendary Member; Posts: 1556; Joined: Tue Aug 14, 2012 11:18 pmFollowed by: 34 members; Thanked: 448 times; GMAT Score:650

by theCodeToGMAT » Mon Jul 14, 2014 5:40 am

y = (x - p)(x - q)

using (2,0)
0 = (2-p)(2-q)
0 = 4 - 2q -2p + pq
2(p+q) = 4 + pq

Statement 1:
We need (p+q) info
INSUFFICIENT

Statement 2:
we need pq info
INSUFFICIENT

Combining...
we have both pq & p+q
SUFFICIENT

[spoiler]{C}[/spoiler]

R A H U L

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Mon Jul 14, 2014 7:15 am

prernamalhotra 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

Thank you,
Prerna

Substitute (2, 0) into y = (x - p)(x - q):
0 = (2 - p) (2 - q).

The equation above is valid if p=2 or q=2.
Question rephrased: Is either p or q equal to 2?

Statement 1: pq = -8
Case 1: p=2 and q=-4
In this case, p is equal to 2.
Case 2: p=1 and q=-8
In this case, neither p nor q is equal to 2.
INSUFFICIENT.

Statement 2: -2 - p = q
Thus:
p+q = -2.
Case 1 also satisfies statement 2.
In Case 1, p is equal to 2.
Case 3: p=1 and q=-3
In this case, neither p nor q is equal to 2.
INSUFFICIENT.

Statements combined:
When the GMAT gives the product of two variables as well as their sum, the two variables are almost certain to be INTEGER VALUES.
Statement 1 implies the following integer values:
p=1, q=-8
p=2, q=-4
p=4, q=-2
p=8, q=-1
p=-1, q=8
p=-2, q=4
p=-4, q=2
p=-8, q=1.
Only the options in red also satisfy statement 2.
In both cases, either p or q is equal to 2.
SUFFICIENT.

The correct answer is C.

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by [email protected] » Mon Jul 14, 2014 10:53 am

Hi prernamalhotra,

This is a great "concept" question; even though it's framed around graphing rules, it's ultimately about a basic arithmetic concept:

Anything multiplied by 0 equals 0.

We're asked if Y = (X - P)(X - Q) crosses at the point (2,0). This is a YES/NO question.

By plugging in the co-ordinate (2,0), the question actually asks....

Is 0 = (2 - P)(2 - Q)?

This ultimately asks....

Does P = 2 and/or does Q = 2?

Fact 1: PQ = -8

Here, we can TEST VALUES:
If P = 2 and Q = -4, then the answer to the question is YES.
If P = -2 and Q = 4, then the answer to the question is NO.
Fact 1 is INSUFFICIENT

Fact 2: -2 - P = Q

This can be rewritten as

P + Q = -2

Again, we can TEST VALUES:
If P = 2 and Q = -4, then the answer to the question is YES.
If P = 0 and Q = -2, then the answer to the question is NO.
Fact 2 is INSUFFICIENT

Combined, we have:
PQ = -8
P + Q = -2

Here, we have a couple of answers, BUT a consistent result:
If P = 2 and Q = -4, then the answer to the question is YES
If P = -4 and Q = 2, then the answer to the question is YES
Combined, SUFFICIENT

Final Answer:C

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Rich

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GMATinsight: Legendary Member; Posts: 1100; Joined: Sat May 10, 2014 11:34 pmFollowed by: 24 members; Location: New Delhi, India; Thanked: 205 times

by GMATinsight » Mon Jul 14, 2014 10:58 am

prernamalhotra 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

Since, y = (x - p)(x - q) therefore
x = p and x = q are the roots of the graphs
((Real)Roots of the equation are the points where the graph intersects the X-Axis)

therefore the for the graph to pass through (2,0) we have to make sure whether atleast one of p and q is/are equal to 2 or not

Question Rephrased: Is/Are p or/and q = 2? [to be answered as YES or NO]

Statement 1) pq = -8

@ p = 1 and q = -8 answering the question NO
@ p = 2 and q = -4 answering the question YES
Inconsistent answer therefore, INSUFFICIENT

Statement 2) -2 - p = q
@ p = -2 and q = 4 answering the question NO
@ p = -4 and q = 2 answering the question YES
Inconsistent answer therefore, INSUFFICIENT

Combining the two statement
pq = -8 and -2 - p = q
p(-2-p) = -8
2p+p^2 = 8

p^2 + 2P - 8 = 0
p^2 + 4P - 2P - 8 = 0
(p+4) (p-2) = 0

i.e. p = -4 or 2
if p = -4 then q = 2 and
if p = 2 then q = -4

therefore either p or q is definitely 2 Answering the Questions as YES
SUFFICIENT

Answer: Option C

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Matt@VeritasPrep: GMAT Instructor; Posts: 2630; Joined: Wed Sep 12, 2012 3:32 pmFollowed by: 119 members; Location: East Bay all the way; Thanked: 625 times; GMAT Score:780

by Matt@VeritasPrep » Wed Jul 16, 2014 11:38 am

Maybe an easier way of thinking about this:

y = (x-p)(x-q) can be written as y = xÂ² - (p+q)*x + pq

If (x,y) = (2,0) satisfies that equation, we'll have 0 = 2Â² - (p+q)*2 + pq.

So the question simplifies to "Is 0 = 4 - 2(p+q) + pq?"

S1 gives us pq, but not (p+q); INSUFFICIENT.
S2 gives us p+q = -2, but not pq; INSUFFICIENT.

Together, we have pq = -8 and p+q = -2. Plugging those into our equation, we have 0 = 4 - 2(-2) + -8, or 0 = 0. That's true, so the two statements together are SUFFICIENT.

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GMATinsight: Legendary Member; Posts: 1100; Joined: Sat May 10, 2014 11:34 pmFollowed by: 24 members; Location: New Delhi, India; Thanked: 205 times

by GMATinsight » Fri Jul 18, 2014 7:25 am

prernamalhotra 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

Another WOW method

for any quadratic equation y = ax^2 + bx + c can be written as y = (x-p)(x-q) i.e. y = x-(p+q)x+pq
where p and q are the roots of equation (Roots are points where graph intersects x-axis)

where The sum of the roots i.e. p+q = -b/a
and The product of the roots i.e. pq = c/a

Here First statement gives Product of the roots and second statement gives sum of the roots

therefore combining them we know the equation and hence the roots (where graph intersects x-axis)

Therefore Answer Option C

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