Answer :

To determine the value of [tex]\( b \)[/tex] that makes the trinomial [tex]\( x^2 - bx + 100 \)[/tex] a perfect square, we need to understand how a perfect square trinomial is structured.

A perfect square trinomial takes the form [tex]\( (x - a)^2 \)[/tex], which expands to:

[tex]\[ (x - a)^2 = x^2 - 2ax + a^2 \][/tex]

Comparing this form with the given trinomial [tex]\( x^2 - bx + 100 \)[/tex], we can identify:

1. The constant term [tex]\( 100 \)[/tex] must be equal to [tex]\( a^2 \)[/tex].
2. The term [tex]\( bx \)[/tex] in the middle must be equal to [tex]\( 2ax \)[/tex].

Let's break this down step-by-step:

1. Find [tex]\( a \)[/tex]:
Since [tex]\( a^2 = 100 \)[/tex], we calculate [tex]\( a \)[/tex] as the square root of [tex]\( 100 \)[/tex]:
[tex]\[
a = \sqrt{100} = 10
\][/tex]

2. Determine [tex]\( b \)[/tex]:
The middle term of the perfect square trinomial, [tex]\( -2ax \)[/tex], should match [tex]\( -bx \)[/tex]. So we have:
[tex]\[
-b = -2a
\][/tex]

Substituting the value of [tex]\( a \)[/tex], we get:
[tex]\[
b = 2 \times 10 = 20
\][/tex]

Thus, the value of [tex]\( b \)[/tex] that makes [tex]\( x^2 - bx + 100 \)[/tex] a perfect square is [tex]\( 20 \)[/tex].

So, the correct answer is A. 20

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