Answer:
(X + 3)² - 9 + 7n.
Explanation:
To complete the square for the quadratic expression X² + 6x + 7n, follow these steps:
Start with the quadratic expression: X² + 6x + 7n.
To complete the square, you want to add and subtract a constant term inside the parentheses. The constant term you add should be (1/2) times the coefficient of the linear term (6x in this case), squared.
(1/2 * 6)^2 = (3)^2 = 9
Add and subtract 9 inside the parentheses:
X² + 6x + 9 - 9 + 7n
Now, rewrite the expression with the perfect square trinomial:
(X² + 6x + 9) - 9 + 7n
The expression inside the parentheses is now a perfect square trinomial, which factors into:
(X + 3)² - 9 + 7n
So, the expression X² + 6x + 7n, when completed, is equivalent to (X + 3)² - 9 + 7n.