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

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

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05 Mar 2018, 00:09
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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
[Reveal] Spoiler: OA

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

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05 Mar 2018, 02:52
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.

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Save up to $250 on examPAL packages (special for GMAT Club members) Last edited by DavidTutorexamPAL on 05 Mar 2018, 07:45, edited 1 time in total. Director Joined: 14 Nov 2014 Posts: 663 Re: A quadratic equation ax^2 + bx + c = 0 has two integral roots x1 and [#permalink] Show Tags 05 Mar 2018, 07:30 1 This post received KUDOS 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. Then (E) is our answer. 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: 210 Re: A quadratic equation ax^2 + bx + c = 0 has two integral roots x1 and [#permalink] Show Tags 05 Mar 2018, 07:44 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. _________________ David Senior tutor at examPAL Signup for a free GMAT course We won some awards: Join our next webinar (free) Save up to$250 on examPAL packages (special for GMAT Club members)

Re: A quadratic equation ax^2 + bx + c = 0 has two integral roots x1 and   [#permalink] 05 Mar 2018, 07:44
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