To solve the quadratic equation [tex]\( x^2 - 18x + 77 = 0 \)[/tex] by completing the square, we can follow these steps:
1. Identify the quadratic term and the linear coefficient:
The given equation is [tex]\( x^2 - 18x + 77 = 0 \)[/tex]. Here, the quadratic term is [tex]\( x^2 \)[/tex] and the linear coefficient is [tex]\( -18 \)[/tex].
2. Halve the linear coefficient and square the result:
Take the linear coefficient, which is [tex]\(-18\)[/tex], halve it to get [tex]\(-9\)[/tex], and then square it. This gives us:
[tex]\[
\left( \frac{-18}{2} \right)^2 = (-9)^2 = 81
\][/tex]
3. Rewrite the quadratic equation using the completed square:
Express [tex]\( x^2 - 18x + 77 \)[/tex] in the form of a perfect square trinomial by adding and subtracting [tex]\( 81 \)[/tex] inside the equation:
[tex]\[
x^2 - 18x + 77 = (x - 9)^2 - 81 + 77
\][/tex]
Simplify the constants:
[tex]\[
(x - 9)^2 - 4 = 0
\][/tex]
4. Isolate the squared binomial:
Move the constant term to the other side of the equation:
[tex]\[
(x - 9)^2 = 4
\][/tex]
Now, the equation [tex]\((x-9)^2=4\)[/tex] matches option:
(1) [tex]\((x-9)^2=4\)[/tex]
Therefore, the correct step in the process of completing the square for the given equation is option (1).
