Simple problem yet confusing, Looking for an explanation to the solution.

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1vam0

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This question is meant to be solved in under 36 seconds.
In the above question, I initially tried to solve it directly by substituting the options in the place of x as the values are relatively small and hence this would seem to be an easier method to solve.
So substituting option (1) will not work as 1/3 + 3 does not equal to 1
option (2) will not work , as if x raised to 3 is 27, it means x = 3 and hence , x/3 + 3/x is not equal to 1
option (3) is incorrect because you will get 3/0
option (4) again if x raised to 3 is -27, x = -3 , substituting -3 in the question you do not get 1 .. so all the options given are incorrect .. .
however, if we decide to solve the question in the following manner,

that is ,on expanding the question , we get : x^2 + 9 = 3x
hence x^2 -3x + 9 = 0 -------> eq1
( x + 3 )^3 = (x+3) (x^2 -3x + 9) { (a + b)^3 = (a+b) (a^2 -ab + b^2) }
as per eq 1 = 0 , on substituting we get,
(x+3)^3=0
or x = -3
hence x^3 = -27 , which is considered as the solution to this question.

Is there a problem in the way it is solved? how is this method justified?
 
1vam0 said:
View attachment 13604

This question is meant to be solved in under 36 seconds.
In the above question, I initially tried to solve it directly by substituting the options in the place of x as the values are relatively small and hence this would seem to be an easier method to solve.
So substituting option (1) will not work as 1/3 + 3 does not equal to 1
option (2) will not work , as if x raised to 3 is 27, it means x = 3 and hence , x/3 + 3/x is not equal to 1
option (3) is incorrect because you will get 3/0
option (4) again if x raised to 3 is -27, x = -3 , substituting -3 in the question you do not get 1 .. so all the options given are incorrect .. .
however, if we decide to solve the question in the following manner,

that is ,on expanding the question , we get : x^2 + 9 = 3x
hence x^2 -3x + 9 = 0 -------> eq1
( x + 3 )^3 = (x+3) (x^2 -3x + 9) { (a + b)^3 = (a+b) (a^2 -ab + b^2) }
as per eq 1 = 0 , on substituting we get,
(x+3)^3=0
or x = -3
hence x^3 = -27 , which is considered as the solution to this question.

Is there a problem in the way it is solved? how is this method justified?
Click to expand...
But if you put x = -3 in your original expression, you get:

x/3 + 3/x = (-3)/3 + 3/(-3) = -2

does not match your given equation of x/3 + 3/x =1

In fact, none of the given solutions fit the original condition (equation)!

In your work above, there are mistakes in the identities you have used. Those should be:

(a + b)3 = a3 + 3 * a2 * b + 3 * a * b2 + b3

and

a3 + b3 = (a + b) * (a2 - a*b + b2) ........................... edited
 
Last edited by a moderator:
Thank you for the correction,
The identity I posted is incorrect.
isn't it
a^3 + b^3 = (a+b)(a^2 - ab + b^2) ?

if we use the correct one ie

x^3 + 3^3 = (x + 3)*(x^2 - 3x + 9 )
from eq 1 , we get x^2 -3x + 9 = 0

hence x^3 = - 3^3 = -27 hence x = -3 .. however this does not satisfy the initial condition given in the question..
 
Last edited:
1vam0 said:
can you please check this , isn't it -a*b ?
Click to expand...
Yes you are correct ... too early in the morning .... I am still a bit bleary eyed!

The correct identity, as you showed in post #4 is:

a3 + b3 = (a + b) * (a2 - a*b + b2)
 
1vam0 said:
View attachment 13604

This question is meant to be solved in under 36 seconds.
In the above question, I initially tried to solve it directly by substituting the options in the place of x as the values are relatively small and hence this would seem to be an easier method to solve.
So substituting option (1) will not work as 1/3 + 3 does not equal to 1
option (2) will not work , as if x raised to 3 is 27, it means x = 3 and hence , x/3 + 3/x is not equal to 1
option (3) is incorrect because you will get 3/0
option (4) again if x raised to 3 is -27, x = -3 , substituting -3 in the question you do not get 1 .. so all the options given are incorrect .. .
however, if we decide to solve the question in the following manner,

that is ,on expanding the question , we get : x^2 + 9 = 3x
hence x^2 -3x + 9 = 0 -------> eq1
( x + 3 )^3 = (x+3) (x^2 -3x + 9) { (a + b)^3 = (a+b) (a^2 -ab + b^2) }
as per eq 1 = 0 , on substituting we get,
(x+3)^3=0
or x = -3
hence x^3 = -27 , which is considered as the solution to this question.

Is there a problem in the way it is solved? how is this method justified?
Click to expand...
The answer is correct; what's happened is that the solutions to the equation x^2 + 9 = 3x are the two complex cube roots of -27, not the real cube root of -27, which you tried. The long way to find this would be to use the quadratic formula to solve for x and then cube it.
 
Dr.Peterson said:
The answer is correct; what's happened is that the solutions to the equation x^2 + 9 = 3x are the two complex cube roots of -27, not the real cube root of -27, which you tried. The long way to find this would be to use the quadratic formula to solve for x and then cube it.
Click to expand...
I lost track of the fact that the "find" of the problem was x3 - not "x" !

So x3 = -27 is the correct answer.

Tricky one.....
 
I would begin with the given:

[MATH]\frac{x}{3}+\frac{3}{x}=1[/MATH]
Cube both sides:

[MATH]\frac{x^3}{3^3}+3\frac{x^2}{3^2}\frac{3}{x}+3\frac{x}{3}\frac{3^2}{x^2}+\frac{3^3}{x^3}=1[/MATH]
[MATH]\frac{x^3}{27}+3\left(\frac{x}{3}+\frac{3}{x}\right)+\frac{27}{x^3}=1[/MATH]
[MATH]\frac{x^3}{27}+3+\frac{27}{x^3}=1[/MATH]
[MATH]\frac{x^3}{27}+\frac{27}{x^3}=-2[/MATH]
[MATH]x^6+54x^3+27^2=0[/MATH]
[MATH](x^3+27)^2=0[/MATH]
[MATH]x^3=-27[/MATH]
 
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