To find the term that must be added to the equation [tex]\( x^2 - 8x = 3 \)[/tex] to make it into a perfect square, let's work through the steps involved:
First, consider the left-hand side of the equation, which is [tex]\( x^2 - 8x \)[/tex].
1. We want to transform this part into a perfect square trinomial of the form [tex]\( (x - a)^2 \)[/tex], where [tex]\( a \)[/tex] is a constant.
2. Recall that a perfect square trinomial takes the form [tex]\( (x - a)^2 = x^2 - 2ax + a^2 \)[/tex].
Here, our goal is to match the form [tex]\( x^2 - 2ax + a^2 \)[/tex] with [tex]\( x^2 - 8x \)[/tex].
3. To do this, we need to find the value of [tex]\( a \)[/tex] such that [tex]\( -2a = -8 \)[/tex]. Solving for [tex]\( a \)[/tex]:
[tex]\[ -2a = -8 \][/tex]
[tex]\[ a = \frac{8}{2} = 4 \][/tex]
4. Now, substitute [tex]\( a = 4 \)[/tex] into the perfect square trinomial form to determine the term we need to add:
[tex]\[ x^2 - 8x + 4^2 = x^2 - 8x + 16 \][/tex]
Thus, the term that needs to be added to complete the square is [tex]\( 16 \)[/tex].
5. Therefore, when we introduce this term, the equation becomes:
[tex]\[ x^2 - 8x + 16 = 3 + 16 \][/tex]
[tex]\[ (x - 4)^2 = 19 \][/tex]
So, the term that must be added is
[tex]\[ \boxed{16} \][/tex]
From the choices given:
A. 64
B. 32
C. -3
D. 16
The correct answer is:
[tex]\[ D. 16 \][/tex]
