If y2+ 14y + k is a perfect square, then k =

Analyze the question: We need k so that y² + 14y + k is a perfect square, meaning it can be written as (y + a)² for some number a.

Concept: A perfect square trinomial has the form: (y + a)² = y² + 2ay + a²

Solution:

Compare y² + 14y + k with y² + 2ay + a².

• The middle terms must match: 2a = 14, so a = 7
• The constant term is then k = a² = 7² = 49

Check: (y + 7)² = y² + 14y + 49 ✓

Why the other options fail:

• 7 is just half of 14, which is a (not a²). This is the most common trap.
• 196 is 14², which is the wrong number to square.
• 4 and 28 do not come from the completing-the-square rule.

Tip: To complete the square, halve the coefficient of y, then square it: (14 ÷ 2)² = 7² = 49. Remember "half it, then square it" and you can solve any question like this in seconds.