For what value of x between − 4 and 4, inclusive, is the value of x^2

Hi,
we can see from the statement that two terms containing x : x^2 will always be positive and -10x will be positive if x is -ive..
so the equation will have greatest value if x is -ive, and lower the value of x, greater is the equation.
so -4 will give the greatest value..

second way could be to factorize it..
\(x^2 − 10x + 16 = (x-8)(x-2)...\)
(x-8)(x-2) will have the greatest value when x is -ive, as both terms will become -ive, and the product of these would give us a positive value..
ans A
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x^2 − 10x + 16
=x^2 - 8x - 2x + 16
=x(x-8) -2(x-8)
= (x-2)(x-8)
Now we can substitute the values of x
x=-4 , then (-6)*(-12) = 72
x=-2 , then (-4)*(-10)= 40
x= 0 , then (-2)*(-8) = 16
x=-2 , then 0* - 8 = 0
x=4 then 2* - 4 = -8

Answer A

Alternatively we can analyse the original expression x^2 − 10x + 16
The term x^2 will be always positive or zero(when x=0)
So value of x^2 will increase as value of x increases .
To maximize -10x , we need the smallest negative value of x in the range [-4,4] . Thus , -4 should give us that max value for -10x .

Answer A
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IMO

x^2 − 10x + 16
= x^2 − 10x + 25 – 9
=(x-5)^2 – 9

=>( x^2 − 10x + 16)max
<=> (x-5)^2 max
<=> x = -4
*with value of x between − 4 and 4, inclusive

So here we can just substitute the values from the options and check for which one does the value of x the greatest. But why is the method in the official guide showing all the values from -4 to 4 inclusive and then finding out the greatest value .Isn't that method simply lengthy??

Let’s first factor the given quadratic.

x^2 − 10x + 16

(x - 8)(x - 2)

In order to make the expression the greatest, we need (x - 8) and (x - 2) to be either both positive or both negative.

Looking at the answer choices, we see that when x is -4, we have the largest possible product:

-12 x -6 = 96

Answer: A
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It tells us the numbers between -4 and 4 inclusive.

From the question stem we have \(x^2 -10x + 16\)

\(x^2\) will always be positive

-10x, if x is negative will provide a positive integer.

From the above we can just directly substitute in the equation with -4 giving the greatest value.

Answer choice A

Put the options values in the quadratic and try you will get maximum value.

remember square of negative no. always positive and multiplication of negative numbers always positive.

The answer should be A
You can create facors (x-8)(x-2)
substitute x= -12*-6= 96
or simple the value of
\(x^2\) − 10x + 16 will be max when all terms are positive.
so we are left with -4 and -2 from choices
-4 will yield better results.
Hence ans A
Can you put this in a better format. Follow thsi to post your result
https://gmatclub.com/forum/11-rules-for- ... 33935.html
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For such questions, put in the values given in solution into the given equation- \(x^2\)-10x+16

Option (1) putting in -4 will give 72
Option (2) putting in -2 will give 40
Option (3) putting in 0 will give 16
Option (4) putting in 2 will give 0
Option (5) putting in 4 will give 8

the greatest number obtained is in option A (72) when x=-4

Hope it helps!

Hi Nums99,

For future reference, you should post your subject-specific questions in their respective sub-forums. For example, the PS Forum can be found here:

https://gmatclub.com/forum/problem-solving-ps-140/

In addition, there's a pretty good chance that many of the practice questions that you might be interested in have already been posted, so you can search them out (for example, through Google). Here is a post discussing the question you are asking about:

https://gmatclub.com/forum/for-what-val ... 11639.html

GMAT assassins aren't born, they're made,
Rich
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Solve this question conceptually. I am assuming x2 is actually x^2, which is always non-negative for any value of x. 16 is always positive. Hence we need to worry about only -10x. We can convert this part into a positive number if we have x as negative. Therefore we can rule of option C to E. We are left with -4 and -2. -4 will not only convert -10x to a positive number but also increase its magnitude. Hence Option A is the correct answer.
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We see that y = x^2 – 10x + 16 is the graph of an up-opening parabola. We can find the x-intercepts by setting the function equal to 0 and factoring:

x^2 – 10x + 16 = 0

(x – 8)(x - 2) = 0

x = 8 or x = 2

Since the x-intercepts are at 2 and 8, we know that the x-coordinate of the vertex of the parabola (which is a minimum value) is halfway between 2 and 8, which is at x = 5.
For an up-opening parabola, the farther an x-value is from the x-value of the vertex, the greater the value of the function. Thus, from the answer choices given, we see that the x-value that is farthest from x = 5 is -4, and so choice A is correct.

Answer: A
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I tried to apply min ax theory used in differentiation and it doesn’t work in such qns. Just a reminder for Engg students

Asked: For what value of x between − 4 and 4, inclusive, is the value of x^2 − 10x + 16 the greatest?

x^2 − 10x + 16 = (x-5)^2 - 9 = (x-8)(x-2)

The value of x^2 − 10x + 16 is greatest when (x-5)^2 is greatest or |x-5| is greatest.


(A) − 4
x^2 − 10x + 16 = 72
(B) − 2
x^2 − 10x + 16 = 40
(C) 0
x^2 − 10x + 16 = 16
(D) 2
x^2 − 10x + 16 = 0
(E) 4
x^2 − 10x + 16 = -8

IMO A
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ScottTargetTestPrep
My initial reaction to seeing this problem was just to plug in the answer choices. Do you see that strategy in this case as an inefficient use of time? I thought of factoring, but just directly plugging in seemed more straightforward to me. Thank you!

Directly plugging in would also save a lot of time.
You can further save time by realizing that x must be negative.
x^2+16 will always be positive, so we are concerned about -10x. Thus x has to be negative and smaller the x, more will be x^2-10x.
-4 is the least value and our answer.
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Your method works just fine!
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