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# A quadratic equation ax^2 + bx + c = 0 has two integral roots x1 and

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Current Student
Joined: 27 May 2014
Posts: 524
GMAT 1: 730 Q49 V41
A quadratic equation ax^2 + bx + c = 0 has two integral roots x1 and  [#permalink]

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04 Mar 2018, 23:09
1
00:00

Difficulty:

75% (hard)

Question Stats:

35% (00:50) correct 65% (02:15) wrong based on 20 sessions

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A quadratic equation ax^2 + bx + c = 0 has two integral roots x1 and x2. If the square of the sum of the roots is 6 greater than the sum of the squares of the roots, which of the following could be the value of the ordered set (a, b, c)?

I. (-1, 4, -3)
II. (1, 4, 3)
III. (3, -10√3, 9)

A) I Only
B) II Only
C) III Only
D) I and II Only
E) I, II and III Only

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examPAL Representative
Joined: 07 Dec 2017
Posts: 868
A quadratic equation ax^2 + bx + c = 0 has two integral roots x1 and  [#permalink]

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Updated on: 05 Mar 2018, 06:45
saswata4s wrote:
A quadratic equation ax^2 + bx + c = 0 has two integral roots x1 and x2. If the square of the sum of the roots is 6 greater than the sum of the squares of the roots, which of the following could be the value of the ordered set (a, b, c)?

I. (-1, 4, -3)
II. (1, 4, 3)
III. (3, -10√3, 9)

A) I Only
B) II Only
C) III Only
D) I and II Only
E) I, II and III Only

We'll translate our question stem into algebra to see what we need to do.
This is a Precise approach.

We're told that $$(x1+x2)^2 = (x1)^2 + (x2)^2 + 6.$$
This simplifies into $$(x1)^2 + 2(x1*x2) + (x2)^2 = (x1)^2 + (x2)^2 + 6.$$
Canceling out the factors on both sides give $$x1*x2 = 3$$

Using the equation x1,x2 = $$\frac{{-b \pm \sqrt{b^2-4ac}}}{2a}$$,
I. is $$\frac{{-4 \pm \sqrt{16-12}}}{-2}$$ which gives x1 = 3, x2 = 1 and x1*x2 = 3 as required.
I. is $$\frac{{-4 \pm \sqrt{16-12}}}{2}$$ which gives x1 = -3, x2 = -1 and x1*x2 = 3 as required.
III. is $$\frac{{10\sqrt{3} \pm \sqrt{100*3-36*3}}}{6}$$ which gives x1 = √3/3, x2 = 3√3. Since x1,x2 are not integers, III is impossible.

_________________

Originally posted by DavidTutorexamPAL on 05 Mar 2018, 01:52.
Last edited by DavidTutorexamPAL on 05 Mar 2018, 06:45, edited 1 time in total.
Director
Joined: 14 Nov 2014
Posts: 632
Location: India
GMAT 1: 700 Q50 V34
GPA: 3.76
Re: A quadratic equation ax^2 + bx + c = 0 has two integral roots x1 and  [#permalink]

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05 Mar 2018, 06:30
1
DavidTutorexamPAL wrote:
saswata4s wrote:
A quadratic equation ax^2 + bx + c = 0 has two integral roots x1 and x2. If the square of the sum of the roots is 6 greater than the sum of the squares of the roots, which of the following could be the value of the ordered set (a, b, c)?

I. (-1, 4, -3)
II. (1, 4, 3)
III. (3, -10√3, 9)

A) I Only
B) II Only
C) III Only
D) I and II Only
E) I, II and III Only

We'll translate our question stem into algebra to see what we need to do.
This is a Precise approach.

We're told that $$(x1+x2)^2 = (x1)^2 + (x2)^2 + 6.$$
This simplifies into $$(x1)^2 + 2(x1*x2) + (x2)^2 = (x1)^2 + (x2)^2 + 6.$$
Canceling out the factors on both sides give $$x1*x2 = 3$$

Using the equation x1,x2 = $$\frac{{-b \pm \sqrt{b^2-4ac}}}{2a}$$,
I. is $$\frac{{-4 \pm \sqrt{16-12}}}{-2}$$ which gives x1 = 3, x2 = 1 and x1*x2 = 3 as required.
I. is $$\frac{{-4 \pm \sqrt{16-12}}}{2}$$ which gives x1 = -3, x2 = -1 and x1*x2 = 3 as required.
III. is $$\frac{{10\sqrt{3} \pm \sqrt{100*3-36*3}}}{6}$$ which gives x1 = √3/3, x2 = 3√3 and x1*x2 = 3 as required.

Hi Exampal
In question stem , it is telling "A quadratic equation ax^2 + bx + c = 0 has two integral roots x1 and x2." , but in case III roots are not integer .
examPAL Representative
Joined: 07 Dec 2017
Posts: 868
Re: A quadratic equation ax^2 + bx + c = 0 has two integral roots x1 and  [#permalink]

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05 Mar 2018, 06:44
sobby wrote:
Hi Exampal
In question stem , it is telling "A quadratic equation ax^2 + bx + c = 0 has two integral roots x1 and x2." , but in case III roots are not integer .

You're right, my mistake.
Fixed in the original post.

Point of fact, since in quadratic equations x1+x2 = -b/a and since in III b/a is irrational we could have eliminated it without even checking.

--== Message from the GMAT Club Team ==--

THERE IS LIKELY A BETTER DISCUSSION OF THIS EXACT QUESTION.
This discussion does not meet community quality standards. It has been retired.

If you would like to discuss this question please re-post it in the respective forum. Thank you!

To review the GMAT Club's Forums Posting Guidelines, please follow these links: Quantitative | Verbal Please note - we may remove posts that do not follow our posting guidelines. Thank you.

_________________
Re: A quadratic equation ax^2 + bx + c = 0 has two integral roots x1 and &nbs [#permalink] 05 Mar 2018, 06:44
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