To solve the quadratic equation [tex]\(x^2 - 12x - 13 = 0\)[/tex] by completing the square, follow these detailed steps:
1. Rearrange the Equation:
Start by moving the constant term to the other side of the equation:
[tex]\[
x^2 - 12x = 13
\][/tex]
2. Complete the Square:
To complete the square on the left-hand side of this equation, we need to add a constant to both sides so that the left-hand side becomes a perfect square trinomial.
- Take the coefficient of [tex]\(x\)[/tex], which is [tex]\(-12\)[/tex].
- Halve this coefficient: [tex]\(-12 / 2 = -6\)[/tex].
- Square the result: [tex]\((-6)^2 = 36\)[/tex].
Add this square to both sides of the equation:
[tex]\[
x^2 - 12x + 36 = 13 + 36
\][/tex]
3. Simplify Both Sides:
Now simplify both sides of the equation:
[tex]\[
x^2 - 12x + 36 = 49
\][/tex]
The left-hand side is a perfect square trinomial:
[tex]\[
(x - 6)^2 = 49
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
To find the value of [tex]\(x\)[/tex], take the square root of both sides of the equation:
[tex]\[
x - 6 = \pm \sqrt{49}
\][/tex]
Since [tex]\(\sqrt{49} = 7\)[/tex], this gives us two equations:
[tex]\[
x - 6 = 7 \quad \text{or} \quad x - 6 = -7
\][/tex]
5. Find the Two Solutions:
Solve each equation for [tex]\(x\)[/tex]:
[tex]\[
x - 6 = 7 \implies x = 6 + 7 \implies x = 13
\][/tex]
[tex]\[
x - 6 = -7 \implies x = 6 - 7 \implies x = -1
\][/tex]
Therefore, the solutions to the equation [tex]\(x^2 - 12x - 13 = 0\)[/tex] are:
[tex]\[
x = -1 \quad \text{or} \quad x = 13
\][/tex]
Thus, the correct answer is:
A. [tex]\(x = -1\)[/tex] or [tex]\(x = 13\)[/tex]
