When the polynomial 6x^2 - 11x - 10 is factored into the form (ax + b)(cx + d), where a, b… — ACT

When the polynomial $6x^2 - 11x - 10$ is factored into the form $(ax + b)(cx + d)$ , where $a$ , $b$ , $c$ , and $d$ are integers such that $a$ and $c$ are positive and $a > c$ , what is the value of $ad - bc$ ?

Answer: -19

Answer

The value of

ad - bc

-19

The correct answer is

-19

. Factoring the polynomial

6x^2 - 11x - 10

yields

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

. Under the constraints that

a

and

c

are positive and

a > c

, we must have

a = 3

b = 2

c = 2

, and

d = -5

. Evaluating the expression

ad - bc

gives

(3)(-5) - (2)(2) = -15 - 4 = -19

Step-by-Step Solution

Factor the quadratic expression

6x^2 - 11x - 10

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

Factoring by grouping is used to rewrite the quadratic trinomial.

Apply the positive coefficient constraints and

a > c

to identify the constants.

a = 3

b = 2

c = 2

d = -5

Since the leading coefficients must be positive and

a > c

, we assign

a = 3

from the first factor and

c = 2

from the second factor.

Evaluate the expression

ad - bc

-19

Substitute the values of

a

b

c

, and

d

to calculate the final numerical value.

Key Concept

Factoring quadratic trinomials of the form

ax^2 + bx + c

where

a > 1

Question

Answer

Step-by-Step Solution

Key Concept

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