parabolas: y = x2 +ax +b

[kelsey.klaver]

CAT Question: If the graph of y=x2+ax+b passes through the points (m,0) and (n,0), what is the value of n-m?

1) 4b=a2-4
2) b=0

I see this is asking for the width of the parabola where it hits the x-axis, but I'm thrown off by b=0. Does this mean it intersects with the y-axis at point 0? Would you mind explaining what this parabola looks like and how to find the points it hits the x axis (m,0) and (n,0) with the given information? Much appreciated.

Kelsey

[RonPurewal]

Some things are unclear about your existing understanding here.

"- What is your approach to statement 1?
The condition b = 0 doesn't apply to that statement, which you haven't addressed in your post.

"- You wrote:

how to find the points it hits the x axis (m,0) and (n,0) with the given information

This is a very dangerous way to think; you're distorting the goal of the problem.
The problem asks for (m - n), not individually for m and n. In DS, if you're not focused on exactly the right goal, you're generally inviting trouble.
(As an easy example of why this is so, let's say that two friends and I pay a total of $24 for lunch. Then, if a, b, and c are the amounts we spent, we know that a + b + c = 24, but we have no clue what a, b, or c is individually.)

In this problem, trying to find specific values for m and n isn't going to work, so that's a big part of the problem.

[RonPurewal]

I'm thrown off by b=0. Does this mean it intersects with the y-axis at point 0?

Try writing a few equations of this sort, with various values of a, and factoring them. (They'll all be very easy to factor.)
A pattern should emerge quickly. If you don't see it, then post (a) what you did see, and (b) where you're still having trouble.

[cavalli27]

Hi Ron,

I have only used MGMAT guides to study concepts and i feel that despite going through the MGMAT geometry guide, i dont have the concepts cleared to solve the really hard questions on coordinate geometry- not ALWAYS but many times. I took the CAT 1 recently (untimed) and am reviewing all questions again- i guessed this one correctly but dont understand the reasoning. I did not understand MGMATs explanation too-

Heres where im lost:

"Therefore, the equation of the parabola can be written as y = (x – m)(x – n), which can be expanded into y = x2 – mx – nx + mn, or y = x2 + (­–m – n)x + mn. Since this equation must be equal to the equation y = x2 + ax + b, it follows that a = –m – n and b = mn. "

I know how to factor quadratics- btw i just reviewed the chapter on Quadratic Eq. in the Foundations of GMAT math just to make sure i could factor fine and looks like i didnt have problems with factoring - perhaps im missing something conceptually and i know this is about factoring- can you help me figure out how to approach this- perhaps with easier examples?

I have the chapter from the strategy guide opened right in front of me and i understand the section on "Function Graphs & Quadratics" and have solved the questions after this chapter correctly - wish your guide was more thorough with more practice questions in the Geometry guide- perhaps im using an older version- 4th edition.

Ill try questions banks for Geometry - still have to scratch for the code at the back of the book- maybe those q's will help.

Thanks !
AK

Update: Since m and n are the 2 roots, you are reverse solving for the quadratic equation?

X=m & X=n SO (x-m)*(x-n)=0 and then you open this up and try arranging it acc to y=x^2+ax+b ?

Just wanted to check whether its the right approach-- thanks!

[jlucero]

Hi Ron,

I have only used MGMAT guides to study concepts and i feel that despite going through the MGMAT geometry guide, i dont have the concepts cleared to solve the really hard questions on coordinate geometry- not ALWAYS but many times. I took the CAT 1 recently (untimed) and am reviewing all questions again- i guessed this one correctly but dont understand the reasoning. I did not understand MGMATs explanation too-

Heres where im lost:

"Therefore, the equation of the parabola can be written as y = (x – m)(x – n), which can be expanded into y = x2 – mx – nx + mn, or y = x2 + (­–m – n)x + mn. Since this equation must be equal to the equation y = x2 + ax + b, it follows that a = –m – n and b = mn. "

I know how to factor quadratics- btw i just reviewed the chapter on Quadratic Eq. in the Foundations of GMAT math just to make sure i could factor fine and looks like i didnt have problems with factoring - perhaps im missing something conceptually and i know this is about factoring- can you help me figure out how to approach this- perhaps with easier examples?

I have the chapter from the strategy guide opened right in front of me and i understand the section on "Function Graphs & Quadratics" and have solved the questions after this chapter correctly - wish your guide was more thorough with more practice questions in the Geometry guide- perhaps im using an older version- 4th edition.

Ill try questions banks for Geometry - still have to scratch for the code at the back of the book- maybe those q's will help.

Thanks !
AK

Update: Since m and n are the 2 roots, you are reverse solving for the quadratic equation?

X=m & X=n SO (x-m)*(x-n)=0 and then you open this up and try arranging it acc to y=x^2+ax+b ?

Just wanted to check whether its the right approach-- thanks!

That's exactly what the explanation is trying to do. There's several ways to get to the right answer. If you plug the two equations you got (a = -m-n and b=mn) into statement 1, you'll be able to get a value for n-m.

BTW: This is a very complicated question, so don't feel bad if the basics of Geometry don't seem to be allowing you to answer the hardest questions. When you've mastered the Geometry and the Algebra concepts, you'll be able to start attempt some of these 750+ problems, but even if you get some of those questions right, the GMAT will have another hard problem up its sleeve. Don't make your goal to be perfect, make your goal to be smart about which questions to attempt on test day.

[cavalli27]

Thank you, Joe ! I am going to look up some other questions on the forums and maybe ask some more questions there!

Regards
AK

[RonPurewal]

As a GUESSING METHOD for frustrating algebra problems, consider the following:

• Sufficient statements generally emerge from the problem itself, because they're the things that solve the problem. I.e., they don't have to be made up at random by the person who creates the problem.

• Insufficient statements, on the other hand, must be made up essentially at random.

So, a GUESSING METHOD:
If you see something that DOESN'T seem to have been made up at random, that statement is more likely to be sufficient.

As an extreme example of what I'm talking about, consider the quadratic formula (with the ±√(b^2 – 4ac) and all that).
That's a very weird formula. If I was going to sit down and make up something off the top of my head... well, it definitely wouldn't look like that. That's the sort of thing that, if it appears at all, comes FROM a problem.

Consider DS #127 in OG13 (#120 in OG12). Can't reproduce it here.
One look at the second statement should tell you that it must have come from somewhere. That's not the kind of thing the problem writers could have made up at random. (E.g., why is that product in there?) So, if you had five seconds, you'd guess that it's sufficient. And you'd be right.

You can do the same thing with statement 1 here.

Again, this is a GUESSING METHOD. You should obviously try to solve the problem first.