For all real numbers a and b, the custom binary operator \star is defined by a \star b = a… — GMAT

For all real numbers $a$ and $b$ , the custom binary operator $\star$ is defined by $a \star b = a^2 - b^2$ . The function $f$ is defined by $f(x) = x \star 3$ . If $f(x) = 4x - 12$ , what is the sum of all real values of $x$ that satisfy this equation?

4Answer
B
3
C
-4
D
-3
E
12

Answer

The sum of all real values of x that satisfy the equation is 4.

Applying the custom operator definition gives f(x) = x^2 - 9. Setting this equal to 4x - 12 produces the quadratic equation x^2 - 4x + 3 = 0. Factoring yields (x - 1)(x - 3) = 0, which has solutions x = 1 and x = 3. The sum of these solutions is 1 + 3 = 4.

Step-by-Step Solution

Evaluate the custom operator expression f(x) = x * 3.

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

Applying the definition of the custom operator a * b = a^2 - b^2 with a = x and b = 3.

Set f(x) equal to 4x - 12 and rearrange into standard quadratic form.

x^2 - 9 = 4x - 12 => x^2 - 4x + 3 = 0

Equating the function expression to the given linear expression.

Factor the quadratic equation to find the roots.

(x - 1)(x - 3) = 0 => x = 1 or x = 3

Finding the values of x that satisfy the quadratic equation.

Calculate the sum of the roots.

1 + 3 = 4

Summing all real solutions to satisfy the prompt requirement.

Key Concept

Custom Operator Evaluation and Quadratic Equation Solving

Estimated Time:1m 30s

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