- Recognize the expression \(x^2 + 10x + 25\) as a perfect square trinomial.
- Identify the algebraic identity \((x + k)^2 = x^2 + 2kx + k^2\).
- Match the terms \(x^2 + 10x + 25\) with \((x + 5)^2\).
- Conclude that the blank is \(\boxed{5}\).
Explanation
- Completing the Square Analysis
We are given the quadratic equation \(x^2 + 10x + 8 = 0\) and we want to solve it by completing the square. The process starts by isolating the terms containing \(x\) on one side:
\[x^2 + 10x = -8\]
To complete the square for the expression \(x^2 + 10x\), we need to add a constant such that the expression becomes a perfect square of the form \((x + k)^2 = x^2 + 2kx + k^2\). Comparing \(x^2 + 10x\) with \(x^2 + 2kx\), we see that \(2k = 10\), which means \(k = 5\). Thus, the constant we need to add is \(k^2 = 5^2 = 25\). Adding 25 to both sides of the equation yields:
\[x^2 + 10x + 25 = -8 + 25\]
\[x^2 + 10x + 25 = 17\]
The left-hand side, \(x^2 + 10x + 25\), is exactly the expansion of \((x + 5)^2\). Therefore, we can rewrite the equation as:
\[(x + 5)^2 = 17\]
Comparing this to the provided step \((x + \boxed{\phantom{5}})^2 = 17\), we see that the value in the box is 5.
Examples
Completing the square is a powerful technique used not only to solve quadratic equations but also to analyze properties of functions, such as finding the vertex of a parabola. For example, if an engineer is modeling the trajectory of a launched projectile, the equation of the path might be given in standard form \(y = ax^2 + bx + c\). By completing the square, the engineer can transform this into the vertex form \(y = a(x - h)^2 + k\), which immediately reveals the maximum height (\(k\)) reached by the projectile and the horizontal position (\(h\)) where it occurs, helping in critical design and safety calculations.