When the algebraic expression 5x^2 - 13x - 6 is factored completely, it can be written in … — Armed Services Vocational Aptitude Battery (ASVAB)

When the algebraic expression $5x^2 - 13x - 6$ is factored completely, it can be written in the form $(5x + a)(x - b)$ , where $a$ and $b$ are positive integers. What is the value of $a + b$ ?

Cevap: 5 / five

Cevap

Factoring the quadratic

5x^2 - 13x - 6

yields

(5x + 2)(x - 3)

. By comparing this to the target format

(5x + a)(x - b)

, we can identify that

a = 2

and

b = 3

. Adding these together gives the final answer of 5.

Adım Adım Çözüm

Determine the factoring method for the non-monic quadratic expression

5x^2 - 13x - 6

Since the leading coefficient is 5, look for two numbers that multiply to

(5)(-6) = -30

and add to the middle coefficient,

-13

This sets up the factor-by-grouping method.

Find the two numbers.

The numbers are

-15

and

2

(-15) \times 2 = -30

and

-15 + 2 = -13

Rewrite the middle term using these two numbers and factor by grouping.

5x^2 - 15x + 2x - 6 = 5x(x - 3) + 2(x - 3) = (5x + 2)(x - 3)

Grouping common terms extracts the binomial factors.

Match the factored form to the given format

(5x + a)(x - b)

Comparing

(5x + 2)(x - 3)

(5x + a)(x - b)

reveals that

a = 2

and

b = 3

Both

a

and

b

correspond exactly to the positive integer components of the binomials.

Calculate the final required sum,

a + b

2 + 3 = 5

The question specifically asks for the sum of the parameters.

Anahtar Kavram

Factoring non-monic quadratic trinomials

Soru

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