To determine the value of [tex]\( c \)[/tex] in the perfect square trinomial [tex]\( x^2 + 10x + c \)[/tex], we follow these steps:
1. Identify the standard form of a perfect square trinomial:
- The standard form of a perfect square trinomial is [tex]\((x + b)^2 = x^2 + 2bx + b^2\)[/tex].
2. Relate the given trinomial to the standard form:
- The given trinomial is [tex]\( x^2 + 10x + c \)[/tex].
- By comparing [tex]\( x^2 + 10x + c \)[/tex] to [tex]\( x^2 + 2bx + b^2 \)[/tex], we see that [tex]\( 2b \)[/tex] corresponds to the coefficient of the linear term (which is 10 in this case).
3. Determine the value of [tex]\( b \)[/tex]:
- Given [tex]\( 2b = 10 \)[/tex], we solve for [tex]\( b \)[/tex] by dividing both sides by 2:
[tex]\[
b = \frac{10}{2} = 5
\][/tex]
4. Calculate the value of [tex]\( c \)[/tex]:
- In the perfect square trinomial formula [tex]\( x^2 + 2bx + b^2 \)[/tex], the constant term [tex]\( c \)[/tex] is equal to [tex]\( b^2 \)[/tex].
- Substitute [tex]\( b = 5 \)[/tex] to find [tex]\( c \)[/tex]:
[tex]\[
c = b^2 = 5^2 = 25
\][/tex]
Therefore, the value of [tex]\( c \)[/tex] is [tex]\( 25 \)[/tex].
So, [tex]\( c = 25 \)[/tex].
