Question 155016
Let the first integer be x, the next consecutive integer is (x+1).
{{{(x+1)^3 - x^3 = 217}}} Expand this.
{{{(x^3+3x^2+3x+1)-x^3 = 217}}} Simplify.
{{{3x^2+3x+1 = 217}}} Subtract 217 from both sides.
{{{3x^2+3x-216 = 0}}} Factor the trinomial.
{{{3(x^2+x-72) = 0}}}
{{{x^2+x-72 = 0}}}
{{{(x-8)(x+9) = 0}}}
{{{x = 8}}} or {{{x = -9}}}
The two integers are 8 and 9
Check:
{{{9^3-8^3 = 729-512}}}={{{217}}}