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Are the roots of quadratic equations equal?

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Are the roots of quadratic equations equal?  [#permalink]

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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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Originally posted by aa008 on 03 Oct 2018, 22:13.
Last edited by aa008 on 04 Oct 2018, 00:40, edited 1 time in total.
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Re: Are the roots of quadratic equations equal?  [#permalink]

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New post 04 Oct 2018, 00:33
aa008 wrote:
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?'
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Re: Are the roots of quadratic equations equal?  [#permalink]

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New post 04 Oct 2018, 00:41
DavidTutorexamPAL wrote:
aa008 wrote:
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?'




thanks for pointing it out.
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It’s not that I’m so smart, it’s just that I stay with problems longer. -- Albert Einstein

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Re: Are the roots of quadratic equations equal?   [#permalink] 04 Oct 2018, 00:41
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