For all real numbers x and y, the custom operator \Delta is defined by x \Delta y = x^2 - … — GMAT

For all real numbers $x$ and $y$ , the custom operator $\Delta$ is defined by $x \Delta y = x^2 - 2y$ . If $k$ is a real number such that $(3 \Delta k) \Delta 4 = 17$ , what is the sum of all possible values of $k$ ?

A
1
B
2
C
8
9Answer
E
14

Answer

The sum of all possible values of

k

is 9.

Applying the custom operation rule

x \Delta y = x^2 - 2y

sequentially, the inner evaluation gives

3 \Delta k = 9 - 2k

. Applying the operation again to

(9 - 2k) \Delta 4

gives

(9 - 2k)^2 - 8 = 17

, which simplifies to

(9 - 2k)^2 = 25

. Taking the square root gives two valid solutions for

9 - 2k

5

and

-5

. Solving

9 - 2k = 5

yields

k = 2

, and solving

9 - 2k = -5

yields

k = 7

. The sum of these two values is

2 + 7 = 9

Step-by-Step Solution

Evaluate the inner custom operation

3 \Delta k

3 \Delta k = 3^2 - 2k = 9 - 2k

Apply the definition

x \Delta y = x^2 - 2y

with

x = 3

and

y = k

Substitute the inner result into the outer expression

(9 - 2k) \Delta 4 = 17

(9 - 2k)^2 - 2(4) = 17 \implies (9 - 2k)^2 - 8 = 17

Apply the definition

x \Delta y = x^2 - 2y

with

x = 9 - 2k

and

y = 4

Isolate the squared term and solve for

9 - 2k

(9 - 2k)^2 = 25 \implies 9 - 2k = 5

9 - 2k = -5

Adding 8 to both sides gives 25; taking the square root requires considering both positive and negative roots.

Solve each linear equation for

k

and calculate their sum.

Case 1:

9 - 2k = 5 \implies 2k = 4 \implies k = 2

.
Case 2:

9 - 2k = -5 \implies 2k = 14 \implies k = 7

.
Sum =

2 + 7 = 9

Solving both equations yields all possible values for

k

Key Concept

Custom Operator Evaluation and Quadratic Equation Root Extraction

Estimated Time:2m 0s

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