For all real values of x and y, the expression 4x^2 - 9y^2 - 12y - 4 can be factored into … — ACT

For all real values of $x$ and $y$ , the expression $4x^2 - 9y^2 - 12y - 4$ can be factored into the form $(2x + ay + b)(2x - cy - d)$ , where $a, b, c$ , and $d$ are positive integers. What is the value of $a + b + c + d$ ?

Answer: 10

Answer

The value of

a + b + c + d

is 10.

Grouping the

y

terms gives

4x^2 - (9y^2 + 12y + 4)

. Factoring the quadratic within the parentheses gives

4x^2 - (3y + 2)^2

. Applying the difference of squares identity

(u^2 - v^2) = (u + v)(u - v)

leads to

(2x + (3y + 2))(2x - (3y + 2)) = (2x + 3y + 2)(2x - 3y - 2)

. Matching this expression to the template

(2x + ay + b)(2x - cy - d)

reveals that

a = 3

b = 2

c = 3

, and

d = 2

, all of which are positive integers. The sum of these values is

3 + 2 + 3 + 2 = 10

Step-by-Step Solution

Group the terms containing

y

4x^2 - (9y^2 + 12y + 4)

Grouping the terms allows us to identify a perfect square trinomial pattern.

Factor the trinomial inside the parentheses

4x^2 - (3y + 2)^2

The expression

9y^2 + 12y + 4

is a perfect square trinomial of the form

(3y)^2 + 2(3y)(2) + 2^2

Apply the difference of squares identity

(2x + (3y + 2))(2x - (3y + 2))

The expression is in the form

u^2 - v^2

, where

u = 2x

and

v = 3y + 2

. Using

u^2 - v^2 = (u + v)(u - v)

factors the expression.

Simplify the factored binomials by distributing signs

(2x + 3y + 2)(2x - 3y - 2)

Simplifying the expressions removes inner parentheses, allowing comparison with the target template.

Compare with the template

(2x + ay + b)(2x - cy - d)

to find the constants

a = 3

b = 2

c = 3

d = 2

Matching the terms directly gives

ay = 3y \Rightarrow a = 3

b = 2

-cy = -3y \Rightarrow c = 3

, and

-d = -2 \Rightarrow d = 2

Calculate the sum of the positive integers

The question asks for the value of

a + b + c + d

, which is

3 + 2 + 3 + 2 = 10

Key Concept

Factoring polynomials using grouping, perfect square trinomials, and the difference of squares identity

Estimated Time:2m 0s

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