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iwill
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quadratic function 27 Feb 2014, 04:45
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How do u determine whether two quadratic functions intersect or not?
please explain with example.



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quadratic function 28 Feb 2014, 15:06
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iwill
How do u determine whether two quadratic functions intersect or not?
please explain with example.
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Dear iwill,
I'm happy to help. :-)

Quadratics are not a frequent topic on the GMAT, and finding where two intersect is a particularly unlikely thing to see. Nevertheless, I love math, so I'm happy to explain.

I assume you are asking about a situation in which you are given two equations, presumably both in the form:
y = a*(x^2) + b*x + c

If that's the case, the easiest thing to do is to set the two expressions that are equal to y equal to each other. For example:

Problem #1
Quadratic #1: y = (x^2) + 2x - 20
Quadratic #2: y = -(x^2) + 4
Just set the equations equal and solve.

(x^2) + 2x - 20 = -(x^2) + 4
add (x^2) to both sides
2*(x^2) + 2x - 20 = 4
subtract 4 from both sides
2*(x^2) + 2x - 24 = 0
divide by 2
(x^2) + x - 12 = 0
(x + 4)(x - 3) = 0

x = -4 and x = +3. The two parabolas intersect at points (+3, -5) and (-4, -12).

Here's another example.

Problem #2
Quadratic #1: y = (x^2) + 3x + 10
Quadratic #2: y = -(x^2) + 3x - 2
Again, set the equations equal.

(x^2) + 3x + 10 = -(x^2) + 3x - 2
subtract 3x from both sides
(x^2) + 10 = -(x^2) - 2
add (x^2) to both sides
2*(x^2) + 10 = -2
subtract 10 from both sides
2*(x^2) = -12
divide by 2
(x^2) = -6
OK, we have a problem. There is no number on the real number line that, when squared, equals negative six. No number on the number line can be a solution to this. This means that these two parabolas DON'T intersect. Here's the picture:
Attachment:
two non-intersecting parabolas.JPG
two non-intersecting parabolas.JPG [ 42.72 KiB | Viewed 2210 times ]

Those are the basics. I don't know whether this answers your question, or whether you have more questions. If you do, feel free to ask.

Mike :-)
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quadratic function 28 Feb 2014, 15:49
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Hi mike,
I. Wanted to confirn.. This was my first question in real gmat. I had to set each equation equal to the main equation and then use disciriminent formula. This was time consuming. Finding out which equation doesn intersect with the given quadratic function. So was wondering f There's any shortcut to it

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quadratic function 28 Feb 2014, 16:12
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Hi mike,
I. Wanted to confirn.. This was my first question in real gmat. I had to set each equation equal to the main equation and then use disciriminent formula. This was time consuming. Finding out which equation doesn intersect with the given quadratic function. So was wondering f There's any shortcut to it
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Dear iwill,
Hmmm. Setting two quadratic equations equal, then collecting terms and calculating the discriminant --- with all due respect, that doesn't sound as if it would take all that long. I think the only thing that could be shorter is if there is something we can solve by inspection. For example
A: y = (x^2) + 2x + 5
B: y = (x^2) + 2x + 9
Here, B is also higher than A, so they never intersect, and if any downward opening parabola that intersects B must also intersect A.

Does all this make sense?
Mike :-)



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