A binary operation \otimes on the set of real numbers \mathbb{R} is defined by a \otimes b… — JAMB UTME

A binary operation $\otimes$ on the set of real numbers $\mathbb{R}$ is defined by $a \otimes b = a^2 + b^2 - ab$ . If $x \otimes 3 = 19$ and $x > 0$ , find the value of $x$ .

Answer: 5

Answer

The value of

x

5

Applying the operation rule gives

x^2 + 3^2 - 3x = 19

, which simplifies to

x^2 - 3x - 10 = 0

. Factoring this equation yields

(x - 5)(x + 2) = 0

. Since

x

is constrained to be positive (

x > 0

), the unique valid answer is

5

Step-by-Step Solution

Apply the definition of the binary operation to

x \otimes 3

x^2 + 3^2 - 3(x) = x^2 - 3x + 9

Substitute

a = x

and

b = 3

into

a \otimes b = a^2 + b^2 - ab

Equate the result to 19 and rearrange into standard quadratic form

x^2 - 3x - 10 = 0

Subtract 19 from both sides to set the quadratic equation to zero.

Factor the quadratic equation and solve for

x

(x - 5)(x + 2) = 0 \implies x = 5 \text{ or } x = -2

Find two numbers that multiply to

-10

and add up to

-3

Apply the restriction

x > 0

x = 5

Reject the negative solution

x = -2

because

x

must be strictly positive.

Key Concept

Evaluation of Binary Operations and Solving Quadratic Equations

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