To solve the problem, we need to analyze the binary operation defined by \( a * b = a^b \) and the equation given, \( a * 2 = 2 - a \). Let's break this down step-by-step. ### Step 1: Understand the Binary Operation The operation \( a * b \) is defined as \( a^b \). This means that when we see \( a * 2 \), it translates to \( a^2 \). ### Step 2: Set Up the Equation From the problem, we have: \[ a * 2 = 2 - a \] Substituting the definition of the operation, we get: \[ a^2 = 2 - a \] ### Step 3: Rearranging the Equation To solve for \( a \), we need to rearrange the equation: \[ a^2 + a - 2 = 0 \] This is a standard quadratic equation in the form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = 1 \), and \( c = -2 \). ### Step 4: Factor the Quadratic Next, we can factor the quadratic equation: \[ (a + 2)(a - 1) = 0 \] This gives us two possible solutions: 1. \( a + 2 = 0 \) → \( a = -2 \) 2. \( a - 1 = 0 \) → \( a = 1 \) ### Step 5: List the Possible Values Thus, the possible values of \( a \) are: \[ a = 1 \quad \text{and} \quad a = -2 \] ### Step 6: Verify the Solutions To ensure these values are correct, we can substitute them back into the original equation \( a * 2 = 2 - a \): 1. For \( a = 1 \): \[ 1 * 2 = 1^2 = 1 \] \[ 2 - 1 = 1 \] Both sides are equal, so \( a = 1 \) is a valid solution. 2. For \( a = -2 \): \[ -2 * 2 = (-2)^2 = 4 \] \[ 2 - (-2) = 2 + 2 = 4 \] Both sides are equal, so \( a = -2 \) is also a valid solution. ### Step 7: Evaluate the Options Now, let's look at the options provided: - **A. 1, -1**: Incorrect because -1 is not a solution. - **B. 1, 2**: Incorrect because 2 is not a solution. - **C. 2, -2**: Incorrect because 2 is not a solution. - **D. 1, -2**: Correct because both 1 and -2 are valid solutions. ### Summary of the Solution - The binary operation is defined as \( a * b = a^b \). - We set up the equation \( a^2 = 2 - a \) and rearranged it to \( a^2 + a - 2 = 0 \). - Factoring gives us the solutions \( a = 1 \) and \( a = -2 \). - The correct answer is option D: 1, -2. ### Revision Summary - Understand the definition of the binary operation \( a * b = a^b \). - Rearrange the equation to form a standard quadratic equation. - Factor the quadratic to find possible values of \( a \). - Verify solutions by substituting back into the original equation.