If #x-y=3# then #x^3-y^3=#?

2 Answers
May 8, 2016

Answer:

The best we can do is:
#color(white)("XXX")x^3-y^3=3(x^2+xy+y^2)#
There is no single solution

Explanation:

We know that in general
#color(white)("XXX")(x^3-y^3)=(x-y)(x^2+xy+y^2)#
and since
#color(white)("XXX")x-y=3#
this gives us
#color(white)("XXX")(x^3-y^3)=3(x^2+xy+y^2)#

There is no single solution.
The table below shows some of the possible combinations:
enter image source here

May 8, 2016

Answer:

See explanation

Explanation:

To apply maths formatting open and close the maths string with the hash symbol. See https://socratic.org/help/symbols

The edit view of this question is:
enter image source here

Given: x-y=3 x^3-y^3=? #" "# Assumed to be:
#" "x-y=3# .............................(1)
#" "x^3-y^3 = ?#..........................(2)

The given question is in an unexpected and unusual mathematical communication format. !!!!

From equation (1) #y=x-3#

Substitution in (2) gives

#x^3-(x-3)^3=?#........................(3)

Consider just the #(x-3)^3#

#(x-3)(x-3)(x-3)#
#(x-3)(x^2-6x+9)#
#x^3-9x^2+27x-27#

Substituting back into (3)

#x^3-(x^3-9x^2+27x-27)#

#9x^2-27x+27#

Set this to equal #y#

#=> y=9x^2-27x+27#

Factor out the 9
#y=9(x^2-3x+3)#.......................(4)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
From this point on it depends what you wish to do with it

#color(blue)("Determine the vertex")#

Consider the #-3x# in equation (4)

Apply #(-1/2)xx-3 = +3/2#

#color(blue)(x_("vertex")=+3/2)#

By substitution in (4)

#color(blue)(y_("vertex")=9[(3/2)^2-3(3/2)+3] =9[ 3/4] = 27/4)#

#color(blue)("Vertex"->(x,y)->(3/2,27/4)#

enter image source here