To solve the quadratic equation [tex]\( x^2 - 8x + 3 = 0 \)[/tex] by completing the square, follow these steps:
1. Move the constant term to the other side of the equation:
[tex]\[
x^2 - 8x = -3
\][/tex]
2. Complete the square on the left side:
To complete the square, we need to take half of the coefficient of [tex]\( x \)[/tex], square it, and add it to both sides of the equation.
The coefficient of [tex]\( x \)[/tex] is [tex]\(-8\)[/tex], so half of that is [tex]\( -4 \)[/tex], and squaring it gives:
[tex]\[
\left( \frac{-8}{2} \right)^2 = 4^2 = 16
\][/tex]
Add this value to both sides of the equation:
[tex]\[
x^2 - 8x + 16 = -3 + 16
\][/tex]
3. Simplify the equation:
[tex]\[
x^2 - 8x + 16 = 13
\][/tex]
4. Express the left side as a perfect square:
The left side of the equation is a perfect square trinomial:
[tex]\[
(x - 4)^2 = 13
\][/tex]
So, the equation used in the process is:
[tex]\[
(x - 4)^2 = 13
\][/tex]
Hence, the correct option is:
[tex]\[
(x - 4)^2 = 13
\][/tex]
