To complete the equation [tex]\( x^2 - 14x + \ldots = (\ldots)^2 \)[/tex], let's go through the process step by step.
The goal is to transform the quadratic expression into a perfect square trinomial and then express it in the form [tex]\((x - a)^2\)[/tex] or [tex]\((x + a)^2\)[/tex].
1. Take the coefficient of [tex]\(x\)[/tex], which is -14, and divide it by 2:
[tex]\[
\frac{-14}{2} = -7
\][/tex]
2. Square the result obtained from step 1:
[tex]\[
(-7)^2 = 49
\][/tex]
3. Add this squared result to the original quadratic equation:
[tex]\[
x^2 - 14x + 49
\][/tex]
4. Now, express the trinomial [tex]\(x^2 - 14x + 49\)[/tex] as a square of a binomial:
[tex]\[
x^2 - 14x + 49 = (x - 7)^2
\][/tex]
Thus, the equation [tex]\( x^2 - 14x + \ldots \)[/tex] is completed as:
[tex]\[
x^2 - 14x + 49 = (x - 7)^2
\][/tex]
Therefore, matching with the given options, we get:
- The term to be added is [tex]\(49\)[/tex]
- The binomial squared is [tex]\((x - 7)^2\)[/tex]
The correct choice is:
B. [tex]\(49 ; x-7\)[/tex]
