If 5 - 6/x = x, then x has how many possible values?

Question

If  5 - \frac{6}{x} = x , then x has how many possible values?

(A) None
(B) One
(C) Two
(D) A finite number greater than two
(E) An infinite number

djuj · Accepted Answer

Since the question ask possible values, Can we apply following rules instead of factoring quadratic equation.
Calculate b^2 and 4ac

If b^2 > 4ac then 2 solutions
If b^2 = 4ac then 1 solution 
If b^2 < 4ac then undefined

I think this method will also handle cases if GMAT choose to provide equations which can't be easily factored.

Bunuel · Answer

If 5 - 6/x = x, then x has how many possible values?

(A) None
(B) One
(C) Two
(D) A finite number greater than two
(E) An infinite number

5 - \frac{6}{x} = x ;

\frac{5x - 6}{x} = x ;

5x -6 = x^2 ;

x^2 - 5x + 6=0 ;

(x-3)(x-2)=0 ;

x=3  or  x=2 .

Answer: C.

akhandamandala · Answer

That's cool Bunuel.
I can approach this kind of equation with Delta method. However I'm very interested in your factoring method.
Would you please share how you quickly factor the equation x^2 - 5x + 6 = 0 into (x-3)(x-2) = 0. Thanks

Bunuel · Answer

Solving and Factoring Quadratics: https://www.purplemath.com/modules/solvquad.htm https://www.purplemath.com/modules/factquad.htm Hope it helps.

akhandamandala · Answer

that's great, it helps a lot. I may save 30 seconds from this method

JeffTargetTestPrep · Answer

We need to simplify the equation 5 – 6/x = x. We start by multiplying the entire equation by x and we obtain:

5x – 6 = x^2

x^2 – 5x + 6 = 0

(x – 3)(x – 2) = 0

x = 3 or x = 2.

We see that x has 2 possible values.

The answer is C.

sisirkant · Answer

As its a quadric equation, there can be 2 values of x. So do you really to calculate or can select answer 2 without calculating what is the value of x?

Bunuel · Answer

Quadratic equation can have 0, 1 or two roots, so you should do some additional math here.

EMPOWERgmatRichC · Answer

Hi All,

You'll have to do a bit of algebra to deduce the number of solutions to this equation…

5 - (6/x) = x

First, multiply everything by x…

5x - 6 = x^2

Now, move everything "to the right"….

0 = x^2 - 5x + 6

You can now factor this into two terms…

0 = (x -2)(x - 3)

And answer the question… There are 2 solutions

Final Answer:  C

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

Siddhuftr · Answer

Well i wanted to reconfirm , we eliminated the possibility of x=0 , because equating x=0 in the equation doesn't solve it , Right ??

Bunuel · Answer

x  is not  0 because after solving the question we got that x = 3 or x = 2 and did not get that x = 0, which naturally means that x = 0 does not satisfy the given equation.

x  cannot be  0 because x is in the denominator of one term, 6/ x , and we cannot divide by 0. If x were 6/x would not be defined.

avigutman · Answer

Video solution from Quant Reasoning: Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1 https://www.youtube.com/watch?v=MjtrVm1zc_A

woohoo921 · Answer

JeffTargetTestPrep Thank you for this helpful reply. To confirm, you cannot just get rid of the x in the denominator of 6 by multiplying it by x on the right side of the equation, right? For example: 5 - (6/x)=x 5-6=x^2 This is incorrect because you would either 1.) need to find like terms on the left side or move the entire (6/x) to the right side by adding it to the right side or moving the 5 to the right side by subtracting it from the right side? I realize this is basic algebra, but that's where my simple mistakes tend to happen

Thank you again.

EMPOWERgmatRichC · Answer

Hi woohoo921,

You CAN multiply both sides of the equation by "X" to remove that fraction - but that means you have to multiply EVERYTHING by X (and in your example, you're only multiplying two of the three pieces of the equation).

With 5 - (6/X) = X

if you multiply both sides by X, you actually have:

5(X) - (6/X)(X) = X(X)
5X - 6 = X^2

You can then move the pieces around and form a quadratic (which you can solve):

X^2 - 5X + 6 = 0
(X - 6)(X + 1) = 0

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

Contact Rich at: [email protected]

AnuK2222 · Answer

hi, was solving this Q on OG website and I was super confused because the variables looked different, I am the only one who saw this weird glitch? (find attached)

Bunuel · Answer

Probably just a typo in your version. Here's how it appears in other versions of the OG. https://gmatclub.com/forum/download/file.php?id=84410 GMAT-Club-Forum-g4udq3vw.png

totaltestprepNick · Answer

C https://www.youtube.com/watch?v=L5hOgO64w-A Nick Slavkovich, GMAT/GRE tutor with 20+ years of experience tutoring@totaltestprep.net

PGTLrowanhand · Answer

Here's my video solution to the problem:  https://www.youtube.com/watch?v=547uoIN8S4k

shahsahil2101 · Answer

This method is good, but its too long.

If 5 - 6/x = x, then x has how many possible values?

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GMAT Problem Solving (PS) Questions

If 5 - 6/x = x, then x has how many possible values?

Prep Toolkit

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

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards