When the expression 4x(x - 2y) - (2x - 3y)^2 + 5y(2x - y) is simplified to the form Ax^2 +… — ACT

When the expression $4x(x - 2y) - (2x - 3y)^2 + 5y(2x - y)$ is simplified to the form $Ax^2 + Bxy + Cy^2$ , where $A$ , $B$ , and $C$ are constants, what is the value of $B$ ?

Answer: 14

Answer

The value of the coefficient

B

is 14.

Expanding the entire expression yields

4x^2 - 8xy - 4x^2 + 12xy - 9y^2 + 10xy - 5y^2

. Grouping and combining the

xy

terms gives

(-8 + 12 + 10)xy = 14xy

. Therefore, the coefficient

B

is 14.

Step-by-Step Solution

Expand the first term

4x^2 - 8xy

Distribute

4x

to both terms inside the parentheses:

4x(x) - 4x(2y) = 4x^2 - 8xy

Expand the squared binomial and apply the negative sign

-4x^2 + 12xy - 9y^2

Use the binomial expansion formula

(2x - 3y)^2 = 4x^2 - 12xy + 9y^2

, then multiply each term by

-1

Expand the third term

10xy - 5y^2

Distribute

5y

to both terms inside the parentheses:

5y(2x) - 5y(y) = 10xy - 5y^2

Combine the coefficients of the like terms

0x^2 + 14xy - 14y^2

Sum the coefficients for each corresponding variable group:

(4 - 4)x^2 + (-8 + 12 + 10)xy + (-9 - 5)y^2

Key Concept

Simplifying Algebraic Expressions and Combining Like Terms

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