To factor the quadratic expression [tex]\( x^2 + 16x + 64 \)[/tex], we can follow these steps:
1. Identify the Coefficients:
The quadratic expression is of the form [tex]\( ax^2 + bx + c \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 16 \)[/tex], and [tex]\( c = 64 \)[/tex].
2. Find the Pair of Numbers:
We need to find two numbers that multiply to [tex]\( ac \)[/tex] and add up to [tex]\( b \)[/tex]. Here, [tex]\( ac = 1 \cdot 64 = 64 \)[/tex] and [tex]\( b = 16 \)[/tex].
- The two numbers that multiply to 64 and add up to 16 are [tex]\( 8 \)[/tex] and [tex]\( 8 \)[/tex].
3. Rewrite the Middle Term:
We rewrite the expression by breaking the middle term (16x) into two terms, using the numbers we found:
[tex]\[
x^2 + 8x + 8x + 64
\][/tex]
4. Group and Factor by Grouping:
Group the terms in pairs and factor out the greatest common factor (GCF) from each pair:
[tex]\[
(x^2 + 8x) + (8x + 64)
\][/tex]
Factor out the GCF from each group:
[tex]\[
x(x + 8) + 8(x + 8)
\][/tex]
5. Factor Out the Common Binomial Factor:
Notice that [tex]\( (x + 8) \)[/tex] is a common binomial factor:
[tex]\[
(x + 8)(x + 8)
\][/tex]
6. Write the Final Factored Form:
The expression can be written as:
[tex]\[
(x + 8)^2
\][/tex]
Therefore, the correct answer is:
C. [tex]\( (x + 8)^2 \)[/tex]
