If 123_x = 38_{10}, what is the value of the base x? — JAMB UTME

If $123_x = 38_{10}$ , what is the value of the base $x$ ?

Answer: 5

The base

x

is 5.

Expanding

123_x

in terms of powers of

x

yields

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

. Setting this equal to the decimal value 38 produces the quadratic equation

x^2 + 2x + 3 = 38

, which simplifies to

x^2 + 2x - 35 = 0

. Factoring gives

(x + 7)(x - 5) = 0

, yielding solutions

x = -7

and

x = 5

. Since a number base must be a positive integer, the correct value for

x

is 5.

Expand the base

x

number into decimal form using place values

123_x = 1 \cdot x^2 + 2 \cdot x^1 + 3 \cdot x^0 = x^2 + 2x + 3

Each digit position in base

x

corresponds to a power of

x

, starting from

x^0

on the right.

Set up and rearrange the quadratic equation

x^2 + 2x + 3 = 38 \implies x^2 + 2x - 35 = 0

Subtracting 38 from both sides converts the equation into standard quadratic form

ax^2 + bx + c = 0

Solve the quadratic equation for

x

(x + 7)(x - 5) = 0 \implies x = -7 \text{ or } x = 5

Factoring gives the roots of the quadratic equation.

Select the valid positive base

x = 5

A base must be a positive integer greater than the largest digit appearing in the number (which is 3).

Place value expansion and base conversion to base 10

Question