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

For all real numbers $a$ and $b$ , the custom binary operation $\star$ is defined by $a \star b = a^2 - 2b^2 + ab$ . The function $g$ is defined by $g(x) = x \star 3$ . What is the positive value of $x$ for which $g(x) = 10$ ?

4Answer
B
1
C
7
D
$\sqrt{28}$
E
3

Answer

The positive value of

x

for which

g(x) = 10

4

Evaluating

g(x) = x \star 3

gives

x^2 - 2(3)^2 + x(3) = x^2 + 3x - 18

. Setting this expression equal to

10

produces the quadratic equation

x^2 + 3x - 28 = 0

. Factoring yields

(x + 7)(x - 4) = 0

, giving solutions

x = -7

and

x = 4

. Since

x

must be positive, the correct value is

4

Step-by-Step Solution

Apply the definition of the custom binary operator

\star

to find

g(x)

g(x) = x \star 3 = x^2 - 2(3)^2 + x(3) = x^2 - 18 + 3x = x^2 + 3x - 18

Substitute

a = x

and

b = 3

into the given formula

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

Set the function expression equal to

10

and rearrange into standard quadratic form.

x^2 + 3x - 18 = 10 \implies x^2 + 3x - 28 = 0

Subtract

10

from both sides to form a quadratic equation equal to zero.

Factor the quadratic equation to find the values of

x

(x + 7)(x - 4) = 0 \implies x = -7 \text{ or } x = 4

Find two numbers that multiply to

-28

and add to

3

, which are

7

and

-4

Select the positive value of

x

as specified in the stem.

x = 4

The question asks specifically for the positive value of

x

Key Concept

Evaluating non-commutative custom operators and solving polynomial equations resulting from function evaluations.

Estimated Time:2m 0s

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