If P = |22x - x2 – 100|, where x is an integer, then what is the maxim

Question

If P = |22x - x^2 – 100|, where x is an integer, then what is the maximum value of x for which P takes the minimum possible value?

A. 6
B. 7
C. 11
D. 15
E. 16

How to solve such questions?

GMATinsight · Accepted Answer

P = |22x - x^2 – 100| = |22x - x^2 – 96 - 4| = |-(x^2 - 22x + 96) - 4| = |-(x-16)*(x-6)) - 4|

@x = 16,  P_{min} = 4

Answer: Option E

nick1816 · Answer

P = |22x - x^2 – 100|

Minimum value of P is 0

-(x^2-22x+100) = 0

x^2-22x+100=0

x= 22/2 ± \sqrt{121-100}

x= 11± √21

We know 4.5^2 = (4*5).25=20.25

√21>4.5

x=11+4.5xyz

x=15.5xyz or 6.4abc

Now minimum integral value of P occurs at the nearest integer value of 15.5xyz or 6.4abc, which are 6 and 16.

At x=6, P =4
At x=16, P=4

Since we have to find maximum value of P, answer must be 16.

Or

a+b=22(equal to sum of roots); We have to choose integer a and b in such a way that product of them is closest to 100.

Use options to choose a and b. Go from the bottom as we need the maximum value

Option E- a=16; b=6; a*b=96 (closest one)
Option D- a=15; b=7; a*b = 105
Option C- a=11; b=11; a*b=121
Don't have to calculate further, as we already did in option E and D

(E)

Kritisood · Answer

Hi, thanks for solution. wanted to understand how did you know to write 100 as 96-4?

mavelous · Answer

He needed to factor the quadratic equation so he knows roots of X where the expression is equal to zero (i.e. minimize P)

Kritisood · Answer

Bunuel   nick1816  is there any theory/generalization on how one can approach such questions? i seem to get lost often with these.

Onyedelmagnifico · Answer

The minimium value of p=0 since the absolute value function yields non negative real numbers. 
Hence at p minimum, 
|22x–x²–100|=0 
Implies 
x²-22x+100=0 
Using the quadratic formula, 
We get 
x=  /2 
x= 11± √21
≈11±5= 16 or 6 
Thus maximum value is 16 (option E)
x=

Posted from my mobile device

GargieMahajan · Answer

Hi, if someone can help me in my equiry:)

in this question 
p=|x^^2-22x+100|
 
if i split manipulate the value ny adding and subtracting 5.
it now looks like |x^2-22x+100+5-5|
 |x^2-22x+105-5|
 |x^2-15x-7x+105-5|
 | (x-7) (x-15) - 5|
 as per this, x=17 and p (min ) should be 5. 
is it a manipulation using answer choices? please suggest me how to appropriate root factors.

Thanks in advance:)
will appreciate help from  egmat

MathRevolution · Answer

P = |22x -  x^2  – 100|

=> For P to be minimum, whole expression |22x -  x^2  – 100| = 0

=>  22x - x^2 – 100 = 0

=>  x^2 - 22x + 100 = 0

We need two numbers whose product is +100 and the addition/subtraction of those same numbers is -22.

Some possible numbers are -15 * - 7 = 105 and -15 -7 = -22. 
=> Other pair is -15 * - 6 = 90 and -15 -6 = -21.

But as we need the product to be 100, our both numbers should be 15.xx and 6.xx and hence the maximum value of 'x' will be 16.

Answer E

Kinshook · Answer

If P = |22x - x^2 – 100|, where x is an integer, then what is the maximum value of x for which P takes the minimum possible value?

P = |x^2 - 22x + 100| = |(x-11)^2 +100 - 121| = |(x-11)^2 - 21|

A. 6 : P = |(6-11)^2-21| = 4
B. 7: P = |(7-11)^2-21| = 5
C. 11 : P = |(11-11)^2-21| = 21
D. 15 : P = |(15-11)^2-21| = 5
E. 16: P = |(16-11)^2-21| = 4

IMO E

If P = |22x - x2 – 100|, where x is an integer, then what is the maxim

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If P = |22x - x2 – 100|, where x is an integer, then what is the maxim

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

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Perfect 805 Score Debrief by Julia

My Rewards