To determine the value of [tex]\( b \)[/tex] that makes the trinomial [tex]\( x^2 - b x + 100 \)[/tex] a perfect square, we can start by recalling the standard form of a perfect square trinomial. It takes the form:
[tex]\[ (x - a)^2 = x^2 - 2ax + a^2 \][/tex]
To match this standard form with the given trinomial [tex]\( x^2 - b x + 100 \)[/tex], let's identify the coefficients:
- The coefficient of [tex]\( x^2 \)[/tex] is 1, which matches.
- The coefficient of [tex]\( x \)[/tex] in the perfect square form is [tex]\( -2a \)[/tex].
- The constant term is [tex]\( a^2 \)[/tex].
Comparing the given trinomial [tex]\( x^2 - b x + 100 \)[/tex] with the perfect square form [tex]\( x^2 - 2ax + a^2 \)[/tex]:
1. The constant term [tex]\( 100 \)[/tex] must be [tex]\( a^2 \)[/tex]:
[tex]\[ a^2 = 100 \][/tex]
2. Solving for [tex]\( a \)[/tex]:
[tex]\[ a = \sqrt{100} \][/tex]
[tex]\[ a = 10 \quad \text{or} \quad a = -10 \][/tex]
Given that [tex]\( a \)[/tex] can be either positive or negative, both values need to be considered.
3. Next, determine the value of [tex]\( b \)[/tex] which is [tex]\( 2a \)[/tex]:
For [tex]\( a = 10 \)[/tex]:
[tex]\[ b = 2 \cdot 10 = 20 \][/tex]
For [tex]\( a = -10 \)[/tex]:
[tex]\[ b = 2 \cdot (-10) = -20 \][/tex]
We are typically interested in the positive value of [tex]\( b \)[/tex] that satisfies the given conditions and matches with the multiple-choice options provided.
Hence, the value of [tex]\( b \)[/tex] that makes the trinomial [tex]\( x^2 - b x + 100 \)[/tex] a perfect square trinomial is:
[tex]\[ \boxed{20} \][/tex]
Thus, The correct answer is B. 20.
