To rewrite the equation [tex]\(x^2 - 2x - 3 = 0 \)[/tex] in the form [tex]\((x - a)^2 = b\)[/tex], follow these steps:
1. Move the constant term to the other side:
[tex]\[
x^2 - 2x = 3
\][/tex]
2. Complete the square:
- Take half of the coefficient of [tex]\(x\)[/tex], which is [tex]\(-2\)[/tex], giving [tex]\(-1\)[/tex].
- Square [tex]\(-1\)[/tex] to get [tex]\(1\)[/tex].
- Add this square to both sides of the equation:
[tex]\[
x^2 - 2x + 1 = 3 + 1
\][/tex]
[tex]\[
x^2 - 2x + 1 = 4
\][/tex]
3. Rewrite the left-hand side as a squared binomial:
[tex]\[
(x - 1)^2 = 4
\][/tex]
From this, we determine that the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are:
[tex]\[
a = 1, \quad b = 4
\][/tex]
Thus, the correct answer is:
[tex]\[
a = 1 \text{ and } b = 4
\][/tex]
