1. **Problem statement:** Given the equation in base $x$: $231_x - 143_x = 44_x$, find the value of $x$. 2. **Convert each number from base $x$ to base 10:** - $231_x = 2x^2 + 3x + 1$ - $143_x = 1x^2 + 4x + 3$ - $44_x = 4x + 4$ 3. **Write the equation in base 10:** $$ (2x^2 + 3x + 1) - (x^2 + 4x + 3) = 4x + 4 $$ 4. **Simplify the left side:** $$ 2x^2 + 3x + 1 - x^2 - 4x - 3 = 4x + 4 $$ $$ (2x^2 - x^2) + (3x - 4x) + (1 - 3) = 4x + 4 $$ $$ x^2 - x - 2 = 4x + 4 $$ 5. **Bring all terms to one side:** $$ x^2 - x - 2 - 4x - 4 = 0 $$ $$ x^2 - 5x - 6 = 0 $$ 6. **Solve the quadratic equation:** Use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a=1$, $b=-5$, $c=-6$. Calculate the discriminant: $$ \Delta = (-5)^2 - 4(1)(-6) = 25 + 24 = 49 $$ Calculate roots: $$ x = \frac{5 \pm 7}{2} $$ Two possible values: - $$ x = \frac{5 + 7}{2} = \frac{12}{2} = 6 $$ - $$ x = \frac{5 - 7}{2} = \frac{-2}{2} = -1 $$ 7. **Interpretation:** Since $x$ is a base, it must be a positive integer greater than any digit in the numbers. The digits are 2, 3, 1, 4, so $x > 4$. Also, $x$ cannot be negative. Therefore, the valid solution is: $$ \boxed{6} $$