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

May 8, 2016

The best we can do is:
$\textcolor{w h i t e}{\text{XXX}} {x}^{3} - {y}^{3} = 3 \left({x}^{2} + x y + {y}^{2}\right)$
There is no single solution

#### Explanation:

We know that in general
$\textcolor{w h i t e}{\text{XXX}} \left({x}^{3} - {y}^{3}\right) = \left(x - y\right) \left({x}^{2} + x y + {y}^{2}\right)$
and since
$\textcolor{w h i t e}{\text{XXX}} x - y = 3$
this gives us
$\textcolor{w h i t e}{\text{XXX}} \left({x}^{3} - {y}^{3}\right) = 3 \left({x}^{2} + x y + {y}^{2}\right)$

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

May 8, 2016

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=? $\text{ }$ Assumed to be:
$\text{ } x - y = 3$ .............................(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 ${\left(x - 3\right)}^{3}$

$\left(x - 3\right) \left(x - 3\right) \left(x - 3\right)$
$\left(x - 3\right) \left({x}^{2} - 6 x + 9\right)$
${x}^{3} - 9 {x}^{2} + 27 x - 27$

Substituting back into (3)

${x}^{3} - \left({x}^{3} - 9 {x}^{2} + 27 x - 27\right)$

$9 {x}^{2} - 27 x + 27$

Set this to equal $y$

$\implies y = 9 {x}^{2} - 27 x + 27$

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

$\textcolor{b l u e}{\text{Determine the vertex}}$

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

Apply $\left(- \frac{1}{2}\right) \times - 3 = + \frac{3}{2}$

$\textcolor{b l u e}{{x}_{\text{vertex}} = + \frac{3}{2}}$

By substitution in (4)

$\textcolor{b l u e}{{y}_{\text{vertex}} = 9 \left[{\left(\frac{3}{2}\right)}^{2} - 3 \left(\frac{3}{2}\right) + 3\right] = 9 \left[\frac{3}{4}\right] = \frac{27}{4}}$

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