To find the equation resulting from completing the square and factoring the given equation [tex]\( x^2 + 16x = 41 \)[/tex], follow these steps:
1. Identify the coefficients: The given equation is [tex]\( x^2 + 16x = 41 \)[/tex].
- Coefficient of [tex]\( x^2 \)[/tex] is [tex]\( 1 \)[/tex].
- Coefficient of [tex]\( x \)[/tex] is [tex]\( 16 \)[/tex].
2. Complete the square:
- First, take the coefficient of [tex]\( x \)[/tex], which is [tex]\( 16 \)[/tex], divide it by [tex]\( 2 \)[/tex], and square the result.
[tex]\[
\left(\frac{16}{2}\right)^2 = 8^2 = 64
\][/tex]
- Add and subtract [tex]\( 64 \)[/tex] inside the equation to complete the square.
[tex]\[
x^2 + 16x + 64 - 64 = 41
\][/tex]
- The left-hand side of the equation can now be written as a perfect square trinomial.
[tex]\[
(x + 8)^2 - 64 = 41
\][/tex]
3. Isolate the squared term:
- Move the constant [tex]\( 64 \)[/tex] to the right-hand side of the equation.
[tex]\[
(x + 8)^2 - 64 + 64 = 41 + 64
\][/tex]
- Simplify the right-hand side.
[tex]\[
(x + 8)^2 = 105
\][/tex]
The equation resulting from completing the square and then factoring is:
[tex]\[
(x + 8)^2 = 105
\][/tex]
Therefore, the correct option is:
C. [tex]\((x+8)^2=105\)[/tex]
