A binary operation x is defined by a x b = a$^b$. If a x 2 = 2 - a, find the possible values of a?

1. Analyze the question

We are given a binary operation defined as $a * b = a^b$. We are also given a specific condition: $a * 2 = 2 - a$. We need to find the values of $a$ that satisfy this equation.

2. Explain the concept

A binary operation is simply a rule for combining two numbers. Here, the rule is "raise the first number to the power of the second number."

• Given: $a * b = a^b$
• Condition: $a * 2 = 2 - a$

We need to substitute $b = 2$ into the definition and solve the resulting algebraic equation.

3. Show the solution

Step 1: Apply the definition to the condition. Since $a * b = a^b$, then $a * 2$ must be $a^2$.

Step 2: Set up the equation. Equate the definition to the given condition: $a^2 = 2 - a$

Step 3: Rearrange into a standard quadratic equation. Move all terms to one side to set the equation to zero: $a^2 + a - 2 = 0$

Step 4: Factorize the quadratic equation. We look for two numbers that multiply to $-2$ and add up to $+1$. These numbers are $+2$ and $-1$. $(a + 2)(a - 1) = 0$

Step 5: Solve for $a$. Set each factor to zero:

• $a + 2 = 0 \implies a = -2$
• $a - 1 = 0 \implies a = 1$

The possible values for $a$ are $1$ and $-2$.

4. Identify the answer

Comparing our results ($1, -2$) with the given options, we see that they match Option A.

5. Tip

When solving quadratic equations in JAMB, always remember the "sum and product" rule. For an equation $ax^2 + bx + c = 0$, the roots must add up to $-b/a$ and multiply to $c/a$. Here, $1 + (-2) = -1$ (which is $-b$) and $1 \times (-2) = -2$ (which is $c$). This is a great way to double-check your work quickly!