If k is a positive constant and the expression 4x^2 + kx + 9 can be written in the form (a… — ACT

If $k$ is a positive constant and the expression $4x^2 + kx + 9$ can be written in the form $(ax + b)^2$ for some integers $a$ and $b$ , what is the value of $k$ ?

A
6
12Answer
C
13
D
24
E
36

Answer

The value of

k

is 12.

A perfect square trinomial is of the form

(ax + b)^2 = a^2x^2 + 2abx + b^2

. Comparing this to

4x^2 + kx + 9

, we have

a^2 = 4

and

b^2 = 9

. Taking the positive roots,

a = 2

and

b = 3

. The coefficient of the middle term is

k = 2ab = 2(2)(3) = 12

. Since

k

is a positive constant, the correct value is 12.

Step-by-Step Solution

Identify the standard form of a perfect square trinomial.

A perfect square trinomial can be written as

(ax + b)^2 = a^2x^2 + 2abx + b^2

This allows us to equate the coefficients of the given expression

4x^2 + kx + 9

to the expanded form.

Solve for the values of

|a|

and

|b|

by equating the coefficients of

x^2

and the constant term.

a^2 = 4 \implies |a| = 2

and

b^2 = 9 \implies |b| = 3

The square of the first term's coefficient is

a^2

and the square of the last term's coefficient is

b^2

Calculate the middle term coefficient

k = 2ab

using the positive values since

k

is a positive constant.

k = 2 \times 2 \times 3 = 12

The middle term of

(ax + b)^2

2abx

, so the coefficient

k

must be

2ab

Key Concept

Factoring perfect square trinomials

Estimated Time:1m 0s

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