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Rewrite [tex]\(x^2 - 2x - 3 = 0\)[/tex] in the form [tex]\((x-a)^2 = b\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are integers, to determine the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex].

A. [tex]\(a = 4\)[/tex] and [tex]\(b = 3\)[/tex]
B. [tex]\(a = 3\)[/tex] and [tex]\(b = 2\)[/tex]
C. [tex]\(a = 2\)[/tex] and [tex]\(b = 1\)[/tex]
D. [tex]\(a = 1\)[/tex] and [tex]\(b = 4\)[/tex]

Sagot :

GinnyAnswer
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]
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