To determine the value that should be added to the expression [tex]\( x^2 - 10x + \square \)[/tex] to make it a perfect-square trinomial, we need to follow these steps:
1. Recognize the Form of a Perfect-Square Trinomial: A perfect-square trinomial can be written as [tex]\( (x - a)^2 \)[/tex], which expands to:
[tex]\[
(x - a)^2 = x^2 - 2ax + a^2
\][/tex]
2. Compare with the Given Expression: The given expression is [tex]\( x^2 - 10x + \square \)[/tex]. This expression will match the form [tex]\( x^2 - 2ax + a^2 \)[/tex] where [tex]\( -2a \)[/tex] corresponds to the coefficient of [tex]\( x \)[/tex], which is [tex]\(-10\)[/tex].
3. Solve for [tex]\( a \)[/tex]: Set up the equation based on the comparison:
[tex]\[
-2a = -10
\][/tex]
Solving for [tex]\( a \)[/tex]:
[tex]\[
2a = 10 \implies a = 5
\][/tex]
4. Find the Missing Term: The missing term that completes the perfect-square trinomial is [tex]\( a^2 \)[/tex]. Calculate [tex]\( a^2 \)[/tex]:
[tex]\[
a^2 = 5^2 = 25
\][/tex]
Therefore, the value that makes the expression [tex]\( x^2 - 10x + \square \)[/tex] a perfect-square trinomial is [tex]\( 25 \)[/tex].
So, the complete perfect-square trinomial is:
[tex]\[
x^2 - 10x + 25 = (x - 5)^2
\][/tex]
