Given following equation: \((142)_𝑏+(112)_{𝑏−2}=(75)_8\), find base \( 𝑏\).

Final answer: b = 5.

Step 1: Convert the right-hand side to decimal.

(75)_8 = 7*8 + 5 = 61.

Step 2: Express the left-hand side in terms of b.

• (142)_b = b^2 + 4b + 2

• (112)_{b-2} = (b-2)^2 + (b-2) + 2

Step 3: Add the two expressions and set equal to 61.

b^2 + 4b + 2 + (b-2)^2 + (b-2) + 2 = 61

Simplify: 2b^2 + b + 6 = 61 → 2b^2 + b - 55 = 0

Step 4: Solve the quadratic.

• Discriminant = 1 + 4*2*55 = 1 + 440 = 441, so √D = 21.

• b = (-1 ± 21) / 4 gives b = 5 or b = -5.5. Discard the negative value, so b = 5.

Step 5: Check digit validity.

For b = 5, (142)_b has digits 1, 4, 2 which are all less than 5. Also b-2 = 3, and (112)_{3} has digits 1, 1, 2 which are all less than 3. So both representations are valid.

Conclusion: b = 5 satisfies the equation.