If the quadratic expression 6x^2 - 7x - 5 is factored completely into the product of two l… — ACT

If the quadratic expression $6x^2 - 7x - 5$ is factored completely into the product of two linear binomials of the form $(ax + b)(cx + d)$ , where $a$ , $b$ , $c$ , and $d$ are integers such that $a > c > 0$ , what is the value of the expression $ad - bc$ ?

Answer: 13

Answer

The value of the expression

ad - bc

is 13.

Step-by-Step Solution

Factor the quadratic expression

6x^2 - 7x - 5

(3x - 5)(2x + 1)

Find two numbers that multiply to

6 \times (-5) = -30

and add to

-7

, which are

-10

and

3

. Rewrite the middle term and factor by grouping:

6x^2 - 10x + 3x - 5 = 2x(3x - 5) + 1(3x - 5) = (3x - 5)(2x + 1)

Determine the values of the coefficients

a

b

c

, and

d

a = 3

b = -5

c = 2

, and

d = 1

The expression is factored into the form

(ax + b)(cx + d)

where

a > c > 0

. Comparing the factors

(3x - 5)

and

(2x + 1)

, we see the coefficients of

x

are

3

and

2

. Since

3 > 2 > 0

, we have

a = 3

and

c = 2

. This leaves

b = -5

and

d = 1

Calculate the value of

ad - bc

Substitute

a = 3

b = -5

c = 2

, and

d = 1

into the expression:

ad - bc = (3)(1) - (-5)(2) = 3 + 10 = 13

Key Concept

Factoring quadratic polynomials of the form

ax^2 + bx + c

with

a > 1

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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