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;\(\left\)(y-5\(\right\))^2$
• B. $25;\(\left\)(y+5\(\right\))^2$
• C. $25;\(\left\)(y-5\(\right\))^2$
• D. $5;\(\left\)(y+5\(\right\))^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 complete the perfect square trinomial.
2. Recall that a perfect square trinomial has the form \(a^2 - 2ab + b^2 = (a - b)^2\). Here, \(a = y\) and the middle term is \(-10y\), which corresponds to \(-2ab\).
3. Set up the equation for the middle term: \(-2ab = -10y\). Since \(a = y\), this means \(-2b = -10\), so solve for \(b\) to find \(b = 5\).
4. Calculate the value to add by squaring \(b\): \(b^2 = 5^2 = 25\). This is the number to add to the expression to make it a perfect square trinomial.
5. Write the completed perfect square trinomial and factor it: \(y^2 - 10y + 25 = (y - 5)^2\).