If 132_x - 54_x = 45_x, where x is a positive integer base, what is the value of x? — JAMB UTME

If $132_x - 54_x = 45_x$ , where $x$ is a positive integer base, what is the value of $x$ ?

The value of the base

x

is 7.

Expanding all numbers in terms of powers of

x

yields

132_x = x^2 + 3x + 2

54_x = 5x + 4

, and

45_x = 4x + 5

. Substituting these into

132_x - 54_x = 45_x

gives

(x^2 + 3x + 2) - (5x + 4) = 4x + 5

. Simplifying this equation results in

x^2 - 6x - 7 = 0

. Factoring

(x - 7)(x + 1) = 0

gives solutions

x = 7

and

x = -1

. Since a base must be a positive integer greater than 5, the base

x

is 7.

Convert each term from base

x

to base 10 using place-value expansion.

132_x = 1 \cdot x^2 + 3 \cdot x^1 + 2 \cdot x^0 = x^2 + 3x + 2

54_x = 5x + 4

, and

45_x = 4x + 5

Converting all terms to a common base (base 10) allows standard algebraic solving.

Substitute the expanded expressions into the given equation.

(x^2 + 3x + 2) - (5x + 4) = 4x + 5

Set up the algebraic equation in terms of

x

Simplify the equation into standard quadratic form.

x^2 - 2x - 2 = 4x + 5 \implies x^2 - 6x - 7 = 0

Combine like terms and move all terms to one side.

Solve the quadratic equation for

x

(x - 7)(x + 1) = 0 \implies x = 7

x = -1

Factor the quadratic expression.

Select the valid base.

x = 7

A number base must be a positive integer greater than the largest digit present in the equation (which is 5).

Solving equations involving unknown number bases by expanding in powers of the base.

Question