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705-805 (Hard)|   Algebra|      
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aa008
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Are the roots of quadratic equations equal? Updated on: 04 Oct 2018, 01:40
Originally posted by aa008 on 03 Oct 2018, 23:13.
Last edited by aa008 on 04 Oct 2018, 01:40, edited 1 time in total.
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Question Stats:

50% (01:59) correct 50% (01:54) wrong based on 155 sessions

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Are the roots of the quadratic equation \(ax^2+bx+c=0\) equal? This quadratic equation has real roots.

1) 2a, b, and 2c are in arithmetic progression.
2) a, b/2, and c are in geometric progression.
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B
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DavidTutorexamPAL
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Are the roots of quadratic equations equal? 04 Oct 2018, 01:33
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aa008
Are the roots of the quadratic equation \(ax^2+bx+c=0\) equal?

1) 2a, b, and 2c are in arithmetic progression.
2) a, b/2, and c are in geometric progression.
Show more

The roots of a quadratic equation are equal only if the expression \(b^2 - 4ac\) equals 0.
We'll look for statements that give us this information, a Logical approach.

(1) This tell us that b = 2a + constant but does not allow us to know if b^2 - 4ac = 0. (example for YES: 2a = b = 2c = 2, a progression with difference 0, example for NO: 2a = 2, b = 10, 2c = 18)
Insufficient.

(2) If \(b/2\) follows \(a\) on an arithemtic progression than \(b/2 = qa\) and \(b = 2qa\). If \(c\) follows \(b/2\) then \(c = qb/2 = q^2a\).
Then \(b^2 - 4ac = (2qa)^2 - 4*a*(q^2a) = 4q^2a^2 - 4q^2a^2 = 0\)
Sufficient.

(B) is our answer.

aa008 Please note that GMAT questions need to be well defined. In this case, it is possible to choose values for a,b,c such that there are no real roots at all. To solve this, you can change to 'assuming that the quadratic equation has real roots, are they equal?'
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Are the roots of quadratic equations equal? 04 Oct 2018, 01:41
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DavidTutorexamPAL
aa008
Are the roots of the quadratic equation \(ax^2+bx+c=0\) equal?

1) 2a, b, and 2c are in arithmetic progression.
2) a, b/2, and c are in geometric progression.

The roots of a quadratic equation are equal only if the expression \(b^2 - 4ac\) equals 0.
We'll look for statements that give us this information, a Logical approach.

(1) This tell us that b = 2a + constant but does not allow us to know if b^2 - 4ac = 0. (example for YES: 2a = b = 2c = 2, a progression with difference 0, example for NO: 2a = 2, b = 10, 2c = 18)
Insufficient.

(2) If \(b/2\) follows \(a\) on an arithemtic progression than \(b/2 = qa\) and \(b = 2qa\). If \(c\) follows \(b/2\) then \(c = qb/2 = q^2a\).
Then \(b^2 - 4ac = (2qa)^2 - 4*a*(q^2a) = 4q^2a^2 - 4q^2a^2 = 0\)
Sufficient.

(B) is our answer.

aa008 Please note that GMAT questions need to be well defined. In this case, it is possible to choose values for a,b,c such that there are no real roots at all. To solve this, you can change to 'assuming that the quadratic equation has real roots, are they equal?'
Show more



thanks for pointing it out.
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TANYA27
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Are the roots of quadratic equations equal? 29 Jul 2021, 19:51
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DavidTutorexamPAL
aa008
Are the roots of the quadratic equation \(ax^2+bx+c=0\) equal?

1) 2a, b, and 2c are in arithmetic progression.
2) a, b/2, and c are in geometric progression.

The roots of a quadratic equation are equal only if the expression \(b^2 - 4ac\) equals 0.
We'll look for statements that give us this information, a Logical approach.

(1) This tell us that b = 2a + constant but does not allow us to know if b^2 - 4ac = 0. (example for YES: 2a = b = 2c = 2, a progression with difference 0, example for NO: 2a = 2, b = 10, 2c = 18)
Insufficient.

(2) If \(b/2\) follows \(a\) on an arithemtic progression than \(b/2 = qa\) and \(b = 2qa\). If \(c\) follows \(b/2\) then \(c = qb/2 = q^2a\).
Then \(b^2 - 4ac = (2qa)^2 - 4*a*(q^2a) = 4q^2a^2 - 4q^2a^2 = 0\)
Sufficient.

(B) is our answer.

aa008 Please note that GMAT questions need to be well defined. In this case, it is possible to choose values for a,b,c such that there are no real roots at all. To solve this, you can change to 'assuming that the quadratic equation has real roots, are they equal?'
Show more
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TANYA27
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Are the roots of quadratic equations equal? 29 Jul 2021, 19:54
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My take on this question...Let me know my thought process is right or wrong.

Statement 1 says that 2a, b, and 2c are in arithmetic progression means that b-2a=2c-b it comes out to be b=c+a we. ant conclude anything from the first statement

From statement 2 it says a, b/2, and c are in geometric progression which means b/2/a=c/b/2 so here it comes out to be b^2=4ac taking the equation on one side it comes out to be b^2-4ac=0 which is our required answer...let me know if my approach is correct or not..Bunuel
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