To determine which value of [tex]\( k \)[/tex] makes the quadratic expression [tex]\( 25x^2 + kx + 4 \)[/tex] a perfect square trinomial, we need to ensure that it can be written in the form [tex]\( (ax + b)^2 \)[/tex].
A perfect square trinomial follows the structure:
[tex]\[ (ax + b)^2 = a^2 x^2 + 2abx + b^2 \][/tex]
Here, the given quadratic equation is:
[tex]\[ 25x^2 + kx + 4 \][/tex]
By comparing this with [tex]\( a^2 x^2 + 2ab x + b^2 \)[/tex]:
1. The coefficient of [tex]\( x^2 \)[/tex] is [tex]\( 25 \)[/tex], so [tex]\( a^2 = 25 \)[/tex].
2. Hence, [tex]\( a = 5 \)[/tex] (since [tex]\( a \)[/tex] is positive).
3. The constant term is [tex]\( 4 \)[/tex], so [tex]\( b^2 = 4 \)[/tex].
4. Therefore, [tex]\( b = 2 \)[/tex] (since [tex]\( b \)[/tex] is positive).
Now, substituting [tex]\( a = 5 \)[/tex] and [tex]\( b = 2 \)[/tex] into the term [tex]\( 2abx \)[/tex]:
[tex]\[ 2ab = 2 \cdot 5 \cdot 2 = 20 \][/tex]
Thus, the value of [tex]\( k \)[/tex] must be [tex]\( 20 \)[/tex] for the quadratic expression [tex]\( 25x^2 + kx + 4 \)[/tex] to be a perfect square trinomial.
Therefore, the correct answer is:
[tex]\[ \boxed{20} \][/tex]
Which corresponds to option D.
