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For integers a and x, which of the following values of a guarantees

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For integers a and x, which of the following values of a guarantees [#permalink]

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For integers \(a\) and \(x\), which of the following values of \(a\) guarantees that \(4x^2 + ax + 16\) is a perfect square?

A. −16
B. −4
C. 4
D. 8
E. 12
[Reveal] Spoiler: OA

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For integers a and x, which of the following values of a guarantees [#permalink]

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New post 23 Nov 2016, 05:51
Bunuel wrote:
For integers \(a\) and \(x\), which of the following values of \(a\) guarantees that \(4x^2 + ax + 16\) is a perfect square?

A. −16
B. −4
C. 4
D. 8
E. 12



IMO A
A would give \(4x^2 - 16x + 16\) which is same as (2x-4)^2
we need to go by the formula \(a^2- 2ab + b^2\)

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For integers a and x, which of the following values of a guarantees [#permalink]

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New post 23 Nov 2016, 06:00
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Bunuel wrote:
For integers \(a\) and \(x\), which of the following values of \(a\) guarantees that \(4x^2 + ax + 16\) is a perfect square?

A. −16
B. −4
C. 4
D. 8
E. 12

Here's another approach:

Given: 4x² + ax + 16

(2x)² = 4x²
4² = 16
So, it appears that EITHER 4x² + ax + 16 = (2x + 4)² OR 4x² + ax + 16 = (2x - 4

Now check each case:
Expand (2x + 4)² to get: 4x² + 16x + 16
Here, a = 16, but 16 is not one of the answer choices

Expand (2x - 4)² to get: 4x² - 16x + 16
Here, a = -16
Answer:
[Reveal] Spoiler:
A


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Re: For integers a and x, which of the following values of a guarantees [#permalink]

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New post 27 Nov 2016, 02:09
I did not get the fundamental of this question. So becasue the two roots of the first answer choice are 2 and 2, it is a perfect square? - please correct me if I am wrong. thank you.

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Re: For integers a and x, which of the following values of a guarantees [#permalink]

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New post 27 Nov 2016, 03:13
Bunuel wrote:
For integers \(a\) and \(x\), which of the following values of \(a\) guarantees that \(4x^2 + ax + 16\) is a perfect square?

A. −16
B. −4
C. 4
D. 8
E. 12


Now, \(4x^2 + ax + 16\) can be of the forms \((2x)^2 - 2(2x)(4) + (4)^2\) = \(4x^2 - 16x + 16\)

Hence, we have -16x = ax

Thus, a = -16

Thus, correct answer will be (A) -16
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Re: For integers a and x, which of the following values of a guarantees [#permalink]

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New post 27 Nov 2016, 22:11
For any quadratic equation ax^2+bx+c = 0 roots are given by
[-b+Root(b^2-4ac)]/2a and [-b-Root(b^2-4ac)]/2a
For roots to be equal b^2-4ac = 0
Applying it here
a=16,-16
From options we get -16 as the answer.

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For integers a and x, which of the following values of a guarantees [#permalink]

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New post 28 Oct 2017, 04:19
OreoShake wrote:
I did not get the fundamental of this question. So becasue the two roots of the first answer choice are 2 and 2, it is a perfect square? - please correct me if I am wrong. thank you.



2 is the root of the equation.Question is whether the equations boils down to a perfect square.
If you put values of x and a in the equation it will result in to zero.Zero is a perfect square.

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Re: For integers a and x, which of the following values of a guarantees [#permalink]

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New post 28 Oct 2017, 19:25
For any quadratic equation ax^2+bx+c = 0 roots
For roots to be equal D, b^2-4ac = 0, distinct if D >0, No roots if D<0
Applying it here
a=16,-16
Checking options, -16 is the answer.
Answer A

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Re: For integers a and x, which of the following values of a guarantees   [#permalink] 28 Oct 2017, 19:25
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