Question

Given that 212basex + 122basex=1111 basex.Find x​

Answer 1

Write out the decimal expansion of each number:

$212_x \equiv 2x^2 + x + 2$

$122_x \equiv x^2 + 2x + 2$

$1111_x \equiv x^3 + x^2 + x + 1$

Note that $x$ must be greater than 2. Solve for $x$.

$212_x + 122_x = 1111_x \iff (2x^2 + x + 2) + (x^2 + 2x + 2) = x^3 + x^2 + x + 1 \\\\ ~~~~ \implies 3x^2 + 3x + 4 = x^3 + x^2 + x + 1 \\\\ ~~~~ \implies x^3 - 2x^2 - 2x - 3 = 0 \\\\ ~~~~ \implies (x^3 - 3x^2) + (x^2 - 3x) + x - 3 = 0 \\\\ ~~~~ \implies x^2(x-3) + x(x-3) + (x-3) = 0 \\\\ ~~~~ \implies (x-3) (x^2+x+1) = 0 \\\\ ~~~~ \implies \boxed{x=3}$