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Intern
Joined: 06 May 2015
Posts: 19
Schools: CUHK '16
GMAT 1: 590 Q47 V24
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12 Jul 2015, 17:56
1
For a quadratic function f(x)=ax^2 + bx + c, we may be aware that, in reality, f(x) can always be greater than zero. In such case, there will be no solution for x.

So far I have not encountered such cases when I am having my revision in GMAT Quantitative session. BUT, will such cases happen in GMAT?

Thanks a lot1
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Joined: 20 Mar 2014
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Schools: Kellogg '18 (M)
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12 Jul 2015, 18:02
1
universehk wrote:
For a quadratic function f(x)=ax^2 + bx + c, we may be aware that, in reality, f(x) can always be greater than zero. In such case, there will be no solution for x.

So far I have not encountered such cases when I am having my revision in GMAT Quantitative session. BUT, will such cases happen in GMAT?

Thanks a lot1

For a quadratic equation to have no REAL solution, the discriminant (D) = $$b^2-4ac$$ needs to be < 0. It is incorrect to say that "f(x) can always be greater than zero. In such case, there will be no solution for x.".

IMO, you are correct that GMAT will not give any quadratic equation that will not have any real values x as this will go against the GMAT's prescribed "all numbers are real numbers" (from Section 5.2, page 150 of Official Guide, 13th Edition).
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14 Jul 2015, 09:56
It is not that there will be no solution... This is an equation and there are actually infinite solutions. What you are saying is that there will be not x-intercepts. That is different than not having a solution.
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Joined: 16 Oct 2010
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15 Jul 2015, 22:14
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universehk wrote:
For a quadratic function f(x)=ax^2 + bx + c, we may be aware that, in reality, f(x) can always be greater than zero. In such case, there will be no solution for x.

So far I have not encountered such cases when I am having my revision in GMAT Quantitative session. BUT, will such cases happen in GMAT?

Thanks a lot1

Understand what is meant by f(x) = ax^2 + bx + c
If we want to depict this equation on the coordinate axis, we say

y = ax^2 + bx + c is an upward sloping parabola if a is positive.

What is the meaning of solving ax^2 + bx + c = 0 for x? It means, when y = 0, what is the value of x? So you are looking for x intercepts.

What is the meaning of solving ax^2 + bx + c = d for x? It means, when y = d, what is the value of x? Depending on the values of a, b, c and d, you may or may not get values for x.

e.g. x^2 - 2x - 3 = 0
(x + 1)(x - 3) = 0
x = -1 or 3

This is what it looks like:
Attachment:

images.jpeg [ 6.03 KiB | Viewed 1081 times ]

So what do you do when you have x^2 - 2x -3 = -3?
You solve it in the same way:
x^2 - 2x -3 + 3 = 0
x(x - 2) = 0
x = 0 or 2

So when y is -3, x is 0 or 2.

Similarly, you can solve for it when y = 5 and get two values for x.

What happens when you put y = -5?
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Director
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Joined: 19 Apr 2009
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17 Jul 2015, 22:51
universehk

Yes, the GMAT does test the concept of a parabola not having any x-intercepts or intersecting the x-axis. Typically the question may come in a data sufficiency format and ask does the parabola y=ax^2+bx+c intersect the x-axis? Most of these problems would fall in the hardest category. Such questions have been tested in the last five years and you will not see them in the Official GMAT guides.

Dabral
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Joined: 09 Sep 2013
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29 Jan 2018, 00:17
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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