The value of x is to be randomly selected from the integers

Question

The value of x is to be randomly selected from the integers from 1 to 10, inclusive, and then substituted into the equation y = x^2 - 4x + 3. What is the probability that the value of y will be negative?

A. 1/10
B. 1/5
C. 1/4
D. 3/10
E. 3/5

A. 1/10
B. 1/5
C. 1/4
D. 3/10
E. 3/5

KarishmaB · Accepted Answer

We need to find that how many of the values of x from 1 to 10 make y negative.

y = x^2 - 4x + 3

We can split the positive and negative terms:
y = x^2 + 3 - 4x
For y to be negative, 4x needs to be greater than (x^2 + 3). We don't need to try x = 4 or greater since in those cases, x^2 + 3 will always be greater than 4x. Think of it this way
6*6 + 3 will be greater than 4*6; 7*7 + 3 will be greater than 4*7 etc

So we just need to figure out x = 1, 2 and 3
For x = 1 and 3, y = 0
For x = 2, y = -1
So only for one value of x, y will be negative. 
Hence required probability = 1/10

Alternatively, you can think of the graph of a quadratic. 
y = x^2 - 4x + 3 = (x - 3)(x - 1) will be a parabola facing upwards with roots at 1 and 3. So it will lie below the x axis only for x = 2.

Bunuel · Answer

Similar questions to practice: if-p-2-13p-40-q-and-p-is-a-positive-integer-between-110590.html new-algebra-set-149349-60.html#p1200970 Hope it helps.

WillEconomistGMAT · Answer

You might try factoring x^2-4x+3 here into...

(x-3)(x-1)

y will be negative when exactly one of these two terms is negative.

Only one value between 1 and 10 will give you that result: 2. So, the answer is A

Temurkhon · Answer

if x^2-4x+3<0
(x-1)*(x-3)<0
x<1, or x<3 
x<1, means that none of 1-10 fits. What is wrong?

LalaB · Answer

y = x^2 - 4x + 3 
y=(x-2)^2-1 in this case y can be negative (-1),if x=2

so, we have only one case out of 10
sure it is 700 level?

Bunuel · Answer

x<1 or x<3 does not make any sense. (x-1)*(x-3)<0 --> 1x2-4x-94661.html#p731476 inequalities-trick-91482.html data-suff-inequalities-109078.html range-for-variable-x-in-a-given-inequality-109468.html everything-is-less-than-zero-108884.html graphic-approach-to-problems-with-inequalities-68037.html Hope this helps.

NateTheGreat11 · Answer

So I get to this point
(x-3)(x-1)
and...
1< P(Y is negative) < 3
But instead I set my equality to less than or equal to, as such:
1 <=P(Y is negative) <= 3
Hence getting 3/10

And I am confused why we don't include the 1 and the 3?
Could you please explain?

Thanks

WillEconomistGMAT · Answer

Notice that a value of 1 either 1 or 3 for x will make y = 0. We are only interested in the negative y values. Hope that helps.

aniketm.87@gmail.com · Answer

We do not include 1 or 3, cuz if we do then the y = 0 y = (x -3)*(x - 1) three ranges : x < 1 1 < x < 3 x > 3 For x < 1, take x = x = 0 y = +ve For x > 3, take x = 5 y = +ve For x = 3 y = (3 - 3)*2 = 0 but for 1 < x < 3, take x = 2 y = -ve, hence only this matches only 1 value out of 10 values (2 out of 1 ,2, 3, 4, ... 10) hence ans is 1/10

daviddaviddavid · Answer

where do I find the difficulty level ?

Bunuel · Answer

The difficulty level is in the tags above the original post:

EMPOWERgmatRichC · Answer

Hi All,

This question requires a certain degree of 'math' to get to the solution, but there's a great built-in logic 'shortcut' that you can take advantage of to minimize the work that you need to do. Look at the equation Y = X^2 - 4X + 3. Let's focus on a specific part of that equation....

X^2 - 4X =

(X)(X) - 4X

When X=4, those two 'pieces' are equal... 
(4)(4) - 4(4) = 0

BUT the extra +3 on the end of the calculation will make that sum POSITIVE. As X gets bigger, the first 'piece' becomes larger at a faster rate than the second pieces does....

X=5 
(5)(5) - 4(5)

X=6 
(6)(6) - 4(6) 
Etc.

So there's really no need to consider ANY number greater than or equal to 4 - we KNOW that the sum will be POSITIVE for all of those values. Instead, we only have to look at the calculations for X=1, X=2 and X=3. That requires some basic 'plug in' arithmetic, then you can answer the question that's asked.

Final Answer:  A

GMAT assassins aren't born, they're made, 
Rich

kapstone1996 · Answer

So my thought process, I missed the question, for trying to be to fast and not diligent

So factored down to

(x-3)(x-1), so should only look at when x = 1,2,3, when X>= 4, each term will be +

So 
X = 1, will get -2*0 = 0, not negative do not count
X = 2, will get -1*1 = -1, do count
x = 3, will get 0*2 = 0, do not count

So only 2 will result in a negative number, thus it is 1/10,

I put one 1/5 because I forgot to consider the 1-1 will result in 0

revamoghe · Answer

Wouldn't it be 4/5? Since except for 1 & 3, any number can have a negative outcome.

EMPOWERgmatRichC · Answer

Hi revamoghe,

To answer this question, we have to consider the possible values for Y (which are based on the 10 possible integer values for X) - and we are asked for the probability that the value of Y will be NEGATIVE. There are a number of different ways to go through that 'math work' (as well as a particular pattern-based shortcut to help you avoid doing most of that work) to figure out the potential values for Y. You will find that there is just ONE value of X that leads to a NEGATIVE value for Y... so the answer to the question is...

1/10.... A 
 
GMAT assassins aren't born, they're made,
Rich

Basshead · Answer

My approach:

y = x^2 - 4x + 3

What is the probability that the value of y will be negative?

Lets plug in some numbers:

5^2 - 4(5) + 3 = 8
4^2 - 4(4) + 3 = 3
3^2 - 4(3) + 3 = 0
 2^2 - 4(2) +3 = -1 
1^2 - 4(1) + 3 = 0

Answer is A. 1/10.

adityaprateek15 · Answer

We want to know for what value of x, y<0

y=x^2-4x+4

x^2-4x+3<0

x^2-3x-x+3<0
(x-3)(x-1)<0

Use the wavy line method using the zero points as 1 and 3, you'll get: 1<x<3

Since x is an integer, the only possible value is 2.

P(selecting #2 out of 10) = 1/10

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The value of x is to be randomly selected from the integers

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