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

Question

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

BrentGMATPrepNow · Accepted Answer

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 )²

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: A

Vinayak Shenoy · Answer

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

OreoShake · Answer

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.

Abhishek009 · Answer

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

sidanand1992 · Answer

For any quadratic equation ax^2+bx+c = 0 roots are given by 
 /2a and  /2a 
For roots to be equal b^2-4ac = 0
Applying it here 
a=16,-16
From options we get -16 as the answer.

nishantt7 · Answer

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.

frndrobinster · Answer

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

ZIX · Answer

Algebraic way:

4x^2 + ax + 16 
 4(x^2 + \frac{(ax)}{4} + 4)

Let the two equal roots be p:

Product of two roots:
 p^2 = 4  --- (1)

Sum of two roots:
 2p = \frac{-a}{4}  (so a is -ve)
 p = \frac{-a}{8}  (substitute in eq 1)

(\frac{-a}{8})^2 = 4

a^2 = 8^2*2^2 
 a = ±16

However, we know a is -ve. So  a = -16

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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

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