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

(1) pq = -8

(2) -2 – p = q

I am not convinced with the explanation given in the MGMAT test.

OA is C.

(1) pq = -8

(2) -2 – p = q

I am not convinced with the explanation given in the MGMAT test.[/quote]

I think I can help you out here:

Y= (X-P)(X-Q) simplifies to:

Y= [X^2-XQ-XP]+PQ

From (1), you know pq = -8, so:

Y= [X^2 - XQ- XP] - 8

Then, you can factor the variables on the right side of the equation in the following manner:

Y= [X(X(-Q-P))] - 8

You know from (2) that -Q-P = 2

So:

Y = [X(X+2)] -8

Now, the question stem asks if the line intercepts (2,0), so plug in 2 for the X value:

Y= [2(2+2)]-8
Y=0

Therefore, YES, it does intercept at 2,0. C is correct.

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

Means that Y=0 when X = p or X = q. or otherwise (p,0) and (q,0) are the points of intersection of the curve with X axis.
(similarly if X=0, then y=pq, which means (0,pq) is the intersection with the Y-axis)

if we want to test for 2,0 then we should be able to prove that either p or q will have a value = 2

(1) pq = -8

Not Sufficient, since we cannot find one single value for p or q (following possibilities with integers, considering these are real numbers will give infinite possibilities)

(p,q)=(-1,8),(-2,4),(-4,2),(-8,1),(1,-8),(2,-4),(4,-2),(8,-1)


(2) -2 – p = q

Cannot use this to prove that p or q = 2


Now use the second equation to find out how many pairs of (p,q) follow the relation. We find that only (-4,2) will fit the relation.

Using both conditions, (-4,0) and 2,0) are the points where the curve intersects the X axis.

perfect explanation above, except for the following:


actually, (p, q) can be either (-4, 2) or (2, -4). i realize that you may know this, but when you write (p, q) = (_, _), the notation implies that the first number is p and the second number is q.

for any readers who don't know, these solutions can be arrived at in one of the following three ways:
(1) (formal method) use statement #2 as a substitution for q, so that pq = -8 gives p(-2 - p) = -8. solve the resulting quadratic for p, and then plug back into q = -2 - p to find q.
(2) (exhaustive listing) just try out all the possibilities for pq = -8, listed by saurav in his post, to see which ones also satisfy -2 - p = q. this may sound daunting at first, but it shouldn't seem that bad when you realize that there are only eight possibilities and the arithmetic is very easy.
(3) (inspection) just stare at the equations and see whether you can think of the numbers that satisfy them. this is a dicey method in general, and, unless your intuition for numbers and algebra is supremely fantastic, should be reserved for situations in which guessing is your only alternative.

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by the way, the most important part of saurav's post is this buried gem:


take note: when a polynomial is written in the form (x - A)(x - B)..., the numbers A, B, ... are the x-intercepts of the polynomial's graph.