The quadratic expression x^2 - 5x - 14 can be factored completely into (x + a)(x + b), whe… — ACT

The quadratic expression $x^2 - 5x - 14$ can be factored completely into $(x + a)(x + b)$ , where $a$ and $b$ are integers and $a > b$ . What is the value of $a$ ?

Answer: 2

Answer

To factor

x^2 - 5x - 14

, we look for two integers that multiply to

-14

and add to

-5

. These integers are

-7

and

2

. Thus, the factored form is

(x - 7)(x + 2)

, which corresponds to

(x + a)(x + b)

where the two constant values are

-7

and

2

. Given the condition

a > b

, the larger value must be assigned to

a

. Since

2 > -7

, we find

a = 2

Step-by-Step Solution

Find two integers that multiply to the constant term

-14

and add to the linear coefficient

-5

The two integers are

-7

and

2

Since

(-7) \times 2 = -14

and

-7 + 2 = -5

, these integers satisfy the requirements for factoring the quadratic trinomial.

Write the quadratic expression in its factored form

(x + p)(x + q)

(x - 7)(x + 2)

The quadratic expression

x^2 + Bx + C

factors into

(x + p)(x + q)

where

p

and

q

are the found integers.

Compare the factored form to the template

(x + a)(x + b)

under the condition

a > b

The set of constants is

\{-7, 2\}

. Since

2 > -7

, we assign

a = 2

and

b = -7

This satisfies the requirement that the integer

a

is strictly greater than the integer

b

Key Concept

Factoring quadratic trinomials with a leading coefficient of 1

Estimated Time:45s

Question

Answer

Step-by-Step Solution

Key Concept

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