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Quadratic equations in DS problems

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nonameee
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Quadratic equations in DS problems [#permalink] New post   19 Nov 2011, 07:05
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Quadratic equations can have two positive solutions. In DS questions, this fact can present an obstacle since in order to decide whether the statement is sufficient we need to know whether the quadratic equation yields one or two positive solutions. Now, to get to a quadratic equation you often have to manipulate/simplify algebraic expressions. This alone can be time consuming. To solve a quadratic equation can take you additional time, especially if the equation doesn't have pretty numbers.

I've got several questions:

1) Is there a quick way to know whether a quadratic equation has one or two positive solutions without the need to solve it?

The only thing that comes to mind is to calculate (or rather to approximate) a discriminant and to compare it with b as in: ax^2 + bx + c = 0

2) Is there a quick way to know whether a quadratic equation has whole solutions?

Here we would have to calculate a discriminant, then to compare it with b (EVEN, ODD).

3) Is there a quick way to know whether the following equation has one positive whole solution:

\(\frac{60}{n}=\frac{60}{n-5}-2\)

I've asked a similar question in this thread:

help-me-on-some-problems-on-og11-ds-99800.html#p1003352

Thanks.
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Re: Quadratic equations in DS problems [#permalink] New post   20 Nov 2011, 21:54
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nonameee wrote:
I've got several questions:

1) Is there a quick way to know whether a quadratic equation has one or two positive solutions without the need to solve it?


Sure. The equation is: \(ax^2 + bx + c = 0\)
We know that sum of the roots = -b/a
Product of the roots = c/a

Provided both the roots are real:
If product of the roots is +ve, either both roots are positive (sum of the roots is positive too) or both roots are negative (sum of the roots is negative)
If product of the roots is -ve, one root is positive and one is negative.

e.g. x^2 + 5x + 4 = 0
Product of roots = 4
Sum of the roots = -5
Both roots negative.

2x^2 - 8x + 1 = 0
Product of roots = 1/2
Sum of roots = 8/2 = 4
Both roots positive.

3x^2 -6x -2 = 0
Product of roots = -2/3
One root negative, one positive

nonameee wrote:
2) Is there a quick way to know whether a quadratic equation has whole solutions?

Here we would have to calculate a discriminant, then to compare it with b (EVEN, ODD).

Yes, you would have to find the roots or use the formula (as you mentioned, find discriminant - it should be a perfect square - sum of its root and b should be even and then divisible by a if a is not 1. Same for the difference.)
nonameee wrote:
3) Is there a quick way to know whether the following equation has one positive whole solution:

\(\frac{60}{n}=\frac{60}{n-5}-2\)


If a whole number solution exists, you can quickly find it by plugging in some clever values. Most of the times (or let's say, almost always), \(\frac{60}{n}\) and \(\frac{60}{n-5}\) will be whole numbers. So I will try n = 6, n = 10, n = 15 etc and get \(\frac{60}{n}\) and \(\frac{60}{n-5}\) as close as possible since their difference must be 2. Most of the times, if a positive whole number solution is there, you will get it this way especially, if it a GMAT-type question. Though, this is more useful in PS questions.
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MichaelH
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Re: Quadratic equations in DS problems [#permalink] New post   20 Nov 2011, 16:52
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nonameee wrote:
1) Is there a quick way to know whether a quadratic equation has one or two positive solutions without the need to solve it?
The only thing that comes to mind is to calculate (or rather to approximate) a discriminant and to compare it with b as in: ax^2 + bx + c = 0


Unless the equation can be solved quickly by inspection, I think that is the only way to know if a quadratic equation has two positive solutions.
For example, it is pretty easy to solve the equation x^2 + 4x - 5 by inspection: (x - 1)(x + 5)
Therefore, the solutions are 1 and -5.

However, if the solution cannot be found quickly by inspection, then you have to calculate the discriminant, D: D = b^2 - 4ac
If D = 0, the equation has two real, equal roots. But you don't know if they are positive without also considering the signs of a and b.

If D <> 0, you have no choice but to compute √D and compare it to b and again taking into consideration the signs of numerator and denominator.

nonameee wrote:
2) Is there a quick way to know whether a quadratic equation has whole solutions?


See my response for #1 above. As far as I know, unless a quick solution can be determined by inspection, you'd have to go through the quadratic formula. (Or draw a quick graph. Or differentiate the equation and see where a maximum occurs and perhaps noting that both zeros are on one side of the y-axis. But I think either of these approaches would be more hassle.)

nonameee wrote:

3) Is there a quick way to know whether the following equation has one positive whole solution:

\(\frac{60}{n}=\frac{60}{n-5}-2\)



Again, there is no quick way to determine this.

You'd have to give all terms in this equation a common denominator and then gather like terms. You'll end up with a quadratic equation:

n^2 - 5n - 150 (assuming I have done it correctly)

Fortunately, this equation can be solved quickly by inspection: (n + 10)(n - 15)

So, yes, this equation has one positive root: 15
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Re: Quadratic equations in DS problems [#permalink] New post   21 Nov 2011, 06:10
Karishma, thanks a lot for your valuable input. Really helpful.
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Re: Quadratic equations in DS problems [#permalink] New post   21 Nov 2011, 15:30
VeritasPrepKarishma wrote:

Sure. The equation is: \(ax^2 + bx + c = 0\)
We know that sum of the roots = -b/a
Product of the roots = c/a

Provided both the roots are real:
If product of the roots is +ve, either both roots are positive (sum of the roots is positive too) or both roots are negative (sum of the roots is negative)
If product of the roots is -ve, one root is positive and one is negative.


This is way helpful. Great information and really useful. For DS to know whether the solutions are positive or negative! And for PS because you can very fast cross out few answer choices with just a calculation. (c/a)
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Re: Quadratic equations in DS problems [#permalink] New post   20 Oct 2021, 11:18
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