When the expression 6x - 2x(x - 3y) - \left(2x - \frac{1}{2}y\right)^2 - \frac{3}{4}x(6x -… — ACT

When the expression $6x - 2x(x - 3y) - \left(2x - \frac{1}{2}y\right)^2 - \frac{3}{4}x(6x - 8)$ is simplified by combining like terms, what is the coefficient of the $x^2$ term?

A
$-\frac{9}{2}$
B
-5
C
-6
D
$-\frac{17}{2}$
$-\frac{21}{2}$ Answer

Answer

-\frac{21}{2}

Expanding the entire expression yields the

x^2

terms

-2x^2

-4x^2

, and

-\frac{9}{2}x^2

. Summing these coefficients gives

-2 - 4 - \frac{9}{2} = -\frac{21}{2}

. Therefore, the coefficient of the

x^2

term is

-\frac{21}{2}

Step-by-Step Solution

Expand the first parenthetical expression by distributing the term

-2x

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

. The expression becomes:

6x - 2x^2 + 6xy - \left(2x - \frac{1}{2}y\right)^2 - \frac{3}{4}x(6x - 8)

Distribution is required to eliminate parentheses before terms can be combined.

Expand the squared binomial

\left(2x - \frac{1}{2}y\right)^2

using the identity

(a - b)^2 = a^2 - 2ab + b^2

, then distribute the negative sign.

\left(2x - \frac{1}{2}y\right)^2 = 4x^2 - 2xy + \frac{1}{4}y^2

. Distributing the negative gives

-4x^2 + 2xy - \frac{1}{4}y^2

This simplifies the second parenthetical term of the expression.

Expand the third parenthetical term by distributing

-\frac{3}{4}x

-\frac{3}{4}x(6x - 8) = -\frac{9}{2}x^2 + 6x

This simplifies the final parenthetical term of the expression.

Combine the coefficients of all the

x^2

terms.

The

x^2

terms are

-2x^2

-4x^2

, and

-\frac{9}{2}x^2

. Combining their coefficients gives:

-2 - 4 - \frac{9}{2} = -6 - \frac{9}{2} = -\frac{12}{2} - \frac{9}{2} = -\frac{21}{2}

Combining like terms simplifies the expression to find the final coefficient of

x^2

Key Concept

Simplifying algebraic expressions by distributing coefficients and combining like terms.

Estimated Time:1m 30s

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