Complete the square for the expression:
$x^2 + 8x + 22$
To complete the square for the quadratic expression \(x^2 + 8x + 22\), we want to write it in the form \(a(x-h)^2 + k\). Here, the coefficient of \(x^2\) is \(1\), so we can focus on the expression \(x^2 + 8x\). To form a perfect square trinomial, we take the coefficient of \(x\), which is \(8\), divide it by \(2\) to get \(4\), and then square it to get \(16\). We add and subtract this value within the expression:
\[x^2 + 8x + 22 = (x^2 + 8x + 16) - 16 + 22.\]
Factoring the perfect square trinomial \((x^2 + 8x + 16)\) gives \((x + 4)^2\), and combining the constants \(-16 + 22\) results in \(6\). Thus, the completed square form is:
\[(x + 4)^2 + 6.\]
Using the completed square form \((x + 4)^2 + 6\), we can now evaluate the expression for the requested values of \(x = 5\) and \(x = 10\).
For \(x = 5\):
\[(5 + 4)^2 + 6 = 9^2 + 6 = 81 + 6 = 87.\]
For \(x = 10\):
\[(10 + 4)^2 + 6 = 14^2 + 6 = 196 + 6 = 202.\]
Completing the square is a vital technique in algebra for solving quadratic equations, finding the vertex of a parabola, or transforming functions. For example, if you are modeling the trajectory of a ball, the vertex form \((x-h)^2 + k\) directly tells you the maximum or minimum height and the time it occurs. By writing \(x^2 + 8x + 22\) as \((x+4)^2 + 6\), we instantly see that the parabola has its vertex at \((-4, 6)\), which is a much clearer insight than the original polynomial form.