Question

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

Answer 1

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:

Answer 2

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:

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)`