During an algebra calculation, an engineer needs to rewrite the expression 2x^2 + 7x - 15 … — Armed Services Vocational Aptitude Battery (ASVAB)

During an algebra calculation, an engineer needs to rewrite the expression $2x^2 + 7x - 15$ in its completely factored form. Which of the following is equivalent to this expression?

$(2x - 3)(x + 5)$ Cevap
B
$(2x + 3)(x - 5)$
C
$(2x - 5)(x + 3)$
D
$(2x + 15)(x - 1)$

Cevap

The correct factorization is

(2x - 3)(x + 5)

The correct answer is mathematically equivalent to the original polynomial. Expanding it using the FOIL method confirms this:

(2x)(x) + (2x)(5) + (-3)(x) + (-3)(5) = 2x^2 + 10x - 3x - 15 = 2x^2 + 7x - 15

Adım Adım Çözüm

Identify the coefficients of the quadratic expression

ax^2 + bx + c

a = 2

b = 7

, and

c = -15

Identifying these values is required to determine the target product and sum for factoring by grouping.

Find two numbers that multiply to

a \times c

and add up to

b

The numbers are

10

and

-3

because

10 \times (-3) = -30

and

10 + (-3) = 7

These specific numbers allow us to logically split the middle term of the polynomial.

Rewrite the middle term

7x

using these two numbers and factor the polynomial by grouping.

2x^2 + 10x - 3x - 15 = 2x(x + 5) - 3(x + 5) = (2x - 3)(x + 5)

Extracting the greatest common factor from each pair reveals the common binomial factor, completing the factorization.

Anahtar Kavram

Factoring non-monic quadratic trinomials by grouping

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