For all real values of x and y, the expression 3x(x - 2y) - 2(x^2 - 4xy + y^2) - x^2 can b… — ACT

For all real values of $x$ and $y$ , the expression $3x(x - 2y) - 2(x^2 - 4xy + y^2) - x^2$ can be written in the form $axy + by^2$ , where $a$ and $b$ are constants. What is the value of $a$ ?

Answer: 2

Answer

The value of

a

is 2.

The value of

a

is 2 because distributing

3x(x - 2y)

yields

3x^2 - 6xy

, and distributing

-2(x^2 - 4xy + y^2)

yields

-2x^2 + 8xy - 2y^2

. Combining these with the

-x^2

term yields

(3-2-1)x^2 + (-6+8)xy - 2y^2 = 2xy - 2y^2

. Comparing this to

axy + by^2

shows that

a

, the coefficient of the

xy

term, is 2.

Step-by-Step Solution

Distribute

3x

across the first parenthetical expression

(x - 2y)

3x^2 - 6xy

To clear the first set of parentheses by multiplying

3x

by each term inside.

Distribute

-2

across the second parenthetical expression

(x^2 - 4xy + y^2)

-2x^2 + 8xy - 2y^2

To clear the second set of parentheses. Note that multiplying

-2

-4xy

yields a positive term

+8xy

due to the sign rules.

Write the full expression and group like terms

(3x^2 - 2x^2 - x^2) + (-6xy + 8xy) - 2y^2

To group terms with identical variable parts so they can be combined.

Combine the coefficients of the grouped terms

2xy - 2y^2

Simplifying the groups:

3-2-1=0

for the

x^2

terms, and

-6+8=2

for the

xy

terms.

Compare the simplified expression to the form

axy + by^2

to find the coefficient

a

a = 2

The coefficient of the

xy

term is

2

, which corresponds to

a

in the target expression.

Key Concept

Simplifying Expressions and Combining Like Terms

Estimated Time:1m 30s

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