"Intercepting X" problem

[am.harel]

The problem reads:
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 explanation says that neither statement alone is sufficient. However, if you plug in statement 1 into the equation, and set x=2, then you do in fact get two sets of possible values for p,q, both of which will intercept the x-axis (y=0). Likewise, the second statement also when plugged in yields two sets of p and q values that both would intercept x-axis. Therefore, I do not understand why the answer is C instead of D.

[stud.jatt]

Lets see. the given equation is

y = (x - p)(x - q); simplifying it we get

y = x^2 - (p+q)x + pq; now substituting pq = -8 we get

y = x^2 -(p+q)x - 8; substituting x = 2 and y = 0 we get

0 = 4 - 2*(p+q) -8; OR

(p+q) = -2

now substituting this value back into the original equation, we get

y = x^2 + 2x - 8;

plug x = 2, y = 0 and VOILA both sides of the equation equal zero

BADA BING, BADA BOOM, solved using statement (1) only


But hold on, not so fast. The correct answer is C not D. Why? Because you made the cardinal mistake of circular reasoning when you first plugged in x=2 and y=0 in the equation y = x^2 -(p+q)x - 8 above. You actually fitted the equation to the desired result. To solve the equation in the right manner, you can only plug the either the value of x and see if that leads to the required value of y (0 here) or vice versa but not both at the same time. And then you'll realize that you have to have the value of (p+q) also to solve it.

How did this happen? Well the only piece of information that statement (1) gives us is that pq = -8 and not the actual values of p and q

for pq = -8, we can have infinite number of values of p and q that satisfy this equation

for ex.
p = -8, q = 1, p*q = -8
p = -4, q = 2, p*q = -8
p = -0.125, q = 64, p*q = -8
p = 1/64000, q = -512000, p*q = -8 and so on

out of these infinite sets is the case when p = -4 and q = 2.

Only for this case does the parabola represented by y = (x - p)(x - q) intercept the x-axis at (2,0)

Hence we have to have the actual values of p and q to answer the question and not just the value of either pq or (p+q) alone. When you combine statement 1 with 2, we get the extra info (p + q) = -2 and are able to answer the question.

Sorry if this is a long post but this type of confusion happens quite often ,that's why I had to draw out your chain of reasoning to point out the main flaw.

The problem reads:
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 explanation says that neither statement alone is sufficient. However, if you plug in statement 1 into the equation, and set x=2, then you do in fact get two sets of possible values for p,q, both of which will intercept the x-axis (y=0). Likewise, the second statement also when plugged in yields two sets of p and q values that both would intercept x-axis. Therefore, I do not understand why the answer is C instead of D.

[jnelson0612]

Wow, excellent! We need to put you on staff here. :-)

[stud.jatt]

Wow, excellent! We need to put you on staff here. :-)

LOL... Thanks for the kind words Jamie; I had applied but was unfortunately rejected. However I still like visiting these forums as they provide an excellent opportunity for mental stimulation.

[jnelson0612]

Wow, excellent! We need to put you on staff here. :-)

LOL... Thanks for the kind words Jamie; I had applied but was unfortunately rejected. However I still like visiting these forums as they provide an excellent opportunity for mental stimulation.

Well, we are VERY happy to have you here giving such excellent explanations. Thanks again! :-)

[jmuduke08]

What triggered you all to know that you can't substitute both values for x and y into this original equation?

[tim]

what makes you think that is not a valid approach to the problem?