If the expression 6x^2 - 11x - 10 is factored completely into the form (ax + b)(cx - d), w… — ACT

If the expression $6x^2 - 11x - 10$ is factored completely into the form $(ax + b)(cx - d)$ , where $a, b, c,$ and $d$ are positive integers, what is the value of $a + b + c + d$ ?

Answer: 12

Answer

The value of the sum of the coefficients and constants

a + b + c + d

12

The factored form of

6x^2 - 11x - 10

(3x + 2)(2x - 5)

. Matching this with

(ax + b)(cx - d)

where

a, b, c,

and

d

are positive integers results in

a = 3

b = 2

c = 2

, and

d = 5

. The sum of these values is

3 + 2 + 2 + 5 = 12

Step-by-Step Solution

Factor the quadratic trinomial

6x^2 - 11x - 10

using the grouping method.

(3x + 2)(2x - 5)

Factoring splits the quadratic expression into its constituent linear binomial factors.

Equate the factored expression to the given form

(ax + b)(cx - d)

to find the values of

a, b, c,

and

d

a = 3

b = 2

c = 2

, and

d = 5

Since the variables represent positive integers, we match the positive constant term to

b

and the negative constant term to

-d

Sum the values of

a, b, c,

and

d

12

Calculating the final sum answers the target mathematical question.

Key Concept

Factoring quadratic polynomials with a leading coefficient greater than 1 using the grouping method.

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