Determine the value we need to add to the equation to make it a perfect square trinomial, then factor it.
$y^2-10y+$__

• A. $5;{(y-5)}^2$
• B. $25;{(y+5)}^2$
• C. $25;{(y-5)}^2$
• D. $5;{(y+5)}^2$

Answer: (C)

Verified step by step guidance

1. Identify the given expression: \(y^2 - 10y + \_\), where we need to find the value to add to make it a perfect square trinomial.
2. Recall that a perfect square trinomial has the form \((y - a)^2 = y^2 - 2ay + a^2\). Here, the middle term is \(-10y\), which corresponds to \(-2ay\).
3. Set \(-2a = -10\) to find \(a\). Solving for \(a\) gives \(a = 5\).
4. Calculate \(a^2 = 5^2 = 25\). This is the value to add to complete the square.
5. Rewrite the trinomial as \(y^2 - 10y + 25\), which factors as \((y - 5)^2\).