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

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Math Expert
Joined: 02 Sep 2009
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For integers a and x, which of the following values of a guarantees  [#permalink]

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23 Nov 2016, 06:28
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Difficulty:

55% (hard)

Question Stats:

54% (01:32) correct 46% (01:12) wrong based on 267 sessions

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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

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

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23 Nov 2016, 06:51
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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23 Nov 2016, 07:00
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Top Contributor
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

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

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27 Nov 2016, 03:09
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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27 Nov 2016, 04:13
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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Joined: 27 Nov 2016
Posts: 1
Re: For integers a and x, which of the following values of a guarantees  [#permalink]

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27 Nov 2016, 23:11
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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28 Oct 2017, 05:19
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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28 Oct 2017, 20:25
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.
Re: For integers a and x, which of the following values of a guarantees &nbs [#permalink] 28 Oct 2017, 20:25
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