The expression a^2+10a+16 can be written is factored form as (a+8)(a+k), where k represents a number. What is the value of k?

Answer 1

We want to factor the quadratic expression \(a^2 + 10a + 16\). We are given that it can be written in the form \((a+8)(a+k)\). To find the value of \(k\), we need to expand the product \((a+8)(a+k)\) and compare it to the original quadratic expression.

First, let's expand \((a+8)(a+k)\):
\[
(a+8)(a+k) = a^2 + ka + 8a + 8k
\]
Combining the like terms, we get:
\[
a^2 + (k+8)a + 8k
\]

Now, we compare this with the original expression \(a^2 + 10a + 16\).

From comparing the terms, we set up the following equations:
\[
k + 8 = 10 \quad \text{(for the coefficient of } a\text{)}
\]
\[
8k = 16 \quad \text{(for the constant term)}
\]

Solving these equations:

1. \(k + 8 = 10\)
\[
k = 10 - 8
\]
\[
k = 2
\]

2. \(8k = 16\)
\[
k = \frac{16}{8}
\]
\[
k = 2
\]

Both equations confirm that \(k = 2\).

Thus, the value of \(k\) is \(2\).