The Correct Answer and Explanation is:
81
The problem asks you to complete the square for the quadratic expression j² + 18j. Completing the square is a technique used to transform a quadratic expression of the form ax² + bx + c into a perfect square trinomial, which is a trinomial that can be factored as (x + k)² or (x – k)².
A perfect square trinomial follows the pattern (x + k)² = x² + 2kx + k². Our goal is to find the constant term, k², that will make the given expression j² + 18j fit this pattern.
In our expression, j takes the place of x. We can compare j² + 18j to the first two terms of the perfect square formula, j² + 2kj.
By comparing the middle terms, 18j and 2kj, we can see that 2k must be equal to 18.
To find the value of k, we solve this simple equation:
2k = 18
k = 18 / 2
k = 9
The number we need to add to complete the square is k². Since we have found that k = 9, we just need to square this value:
k² = 9² = 81
Therefore, the number that completes the square is 81.
When we add 81 to the original expression, we get j² + 18j + 81. This is a perfect square trinomial because it can be factored into (j + 9)². We can verify this by expanding (j + 9)², which gives us (j + 9)(j + 9) = j² + 9j + 9j + 81 = j² + 18j + 81, confirming our result. The procedure provides a reliable method for finding the missing term in any such quadratic expression.
