Choose the value of $c$ to form a perfect-square trinomial.
$x^2+8 x+c$
A perfect-square trinomial is a quadratic expression of the form \(x^2 + 2kx + k^2\), which can be factored as \((x + k)^2\). We are given the expression \(x^2 + 8x + c\) and want to determine the constant \(c\) that makes this a perfect square. This involves identifying the coefficient of the linear term and relating it to the general form of a perfect-square trinomial.
The given expression is \(x^2 + 8x + c\). Comparing this to the general form \(x^2 + 2kx + k^2\), we can set the linear term coefficients equal to each other:
\[2k = 8\]
Solving for \(k\), we divide both sides by 2:
\[k = \frac{8}{2} = 4\]
Now that we have found \(k = 4\), we know that for the expression to be a perfect square, the constant term \(c\) must satisfy \(c = k^2\).
Calculating this, we get:
\[c = 4^2 = 16\]
Therefore, the perfect-square trinomial is \(x^2 + 8x + 16\), which factors as \((x + 4)^2\).
Comparing our result with the given choices:
Our calculated value is \(16\), which corresponds to option a. Thus, the value of \(c\) is \(\boxed{16}\).
Understanding how to complete the square is essential for solving various quadratic equations in real life. For example, if an architect is designing a room and needs to calculate the area of a square floor where the length is related to the width, they might encounter expressions that need to be converted into a perfect square to find the dimensions. This mathematical skill provides a foundation for graphing parabolas and optimizing physical space in design and construction projects.