Question

What value should be added in the blanks to complete the square?

x^2 + 8x + ___ + y^2 + 10y + 25 = -15 + ___ + 25

A) 4
B) 8
C) 16
D) 64

Answer 1

The value that should be added to complete the square for the expression x² + 8x is 16. This value creates a perfect square trinomial, allowing the equation to be balanced when adding the same value to both blanks.

Step-by-step explanation:

To complete the square for the quadratic expression in x, we look for a value that makes x² + 8x into a perfect square trinomial. The process involves taking half of the linear coefficient (which is 8 in this case), squaring it, and adding it to both sides of the equation. The value that completes the square is (8/2)², which is 16. Similarly, for the expression in y, the constant term 25 already completes the square for y² + 10y, since (10/2)² = 25, and this value is included in the original equation.

Now, add this value to both blanks to maintain the balance of the equation: x² + 8x + 16 + y² + 10y + 25 = -15 + 16 + 25. This results in a squared term for x (which is (x + 4)²) and a squared term for y (which is (y + 5)²).