If 214_x = 312_5, where x represents a positive integer base, find the value of x. — JAMB UTME

If $214_x = 312_5$ , where $x$ represents a positive integer base, find the value of $x$ .

Answer: 6

The value of the base

x

is 6.

Expanding

312_5

to base 10 gives

3(25) + 1(5) + 2(1) = 82

. Expanding

214_x

gives

2x^2 + x + 4

. Equating both expressions yields

2x^2 + x - 78 = 0

. Factoring as

(2x + 13)(x - 6) = 0

gives the positive integer root

x = 6

Convert the right side of the equation from base 5 to base 10

312_5 = 3 \times 5^2 + 1 \times 5^1 + 2 \times 5^0 = 75 + 5 + 2 = 82_{10}

Converting known non-decimal bases to base 10 provides a standard baseline for algebraic manipulation.

Expand the left side expression in terms of powers of

x

214_x = 2x^2 + x + 4

Positional values in base

x

correspond to powers of

x

(

x^2, x^1, x^0

Formulate and rearrange the resulting quadratic equation in standard form

2x^2 + x + 4 = 82 \implies 2x^2 + x - 78 = 0

Setting the base 10 expansions equal forms a quadratic equation.

Solve the quadratic equation for positive integer values of

x

x = 6

Factoring

(2x + 13)(x - 6) = 0

gives

x = 6

x = -6.5

. A base must be a positive integer larger than all individual digits appearing in the number (digits are 2, 1, 4, so

x > 4

Converting non-decimal numbers to base 10 using positional expansion to solve polynomial equations involving unknown bases.

Question