If 43_x + 56_x = 121_x, where x represents a positive integer base, find the value of x. — JAMB UTME

If $43_x + 56_x = 121_x$ , where $x$ represents a positive integer base, find the value of $x$ .

Answer: 8

The value of the base

x

is 8.

Expanding each base

x

number into polynomial form gives

(4x + 3) + (5x + 6) = x^2 + 2x + 1

. Simplifying yields the quadratic equation

x^2 - 7x - 8 = 0

, which factors as

(x - 8)(x + 1) = 0

. Since a number base must be a positive integer greater than any digit present in the problem (

x > 6

x = 8

Convert all base

x

numbers into base 10 algebraic expressions.

43_x = 4x + 3

56_x = 5x + 6

, and

121_x = x^2 + 2x + 1

Place-value expansion expresses numbers in base

x

as polynomials in

x

Set up the algebraic equation corresponding to the addition.

(4x + 3) + (5x + 6) = x^2 + 2x + 1 \implies 9x + 9 = x^2 + 2x + 1

The sum of the left-hand terms equals the right-hand term.

Rearrange into standard quadratic form and factor.

x^2 - 7x - 8 = 0 \implies (x - 8)(x + 1) = 0

Moving all terms to one side allows solving for the roots of the quadratic equation.

Determine the valid base value.

x = 8

Number bases must be positive integers greater than all individual digits present in the expression (

x > 6

Unknown base equations and expansion

Question