To complete the square for the equation [tex]\( x^2 - 8x = 9 \)[/tex], follow these steps:
1. Identify the coefficient of [tex]\( x \)[/tex] in the quadratic term [tex]\( x^2 - 8x \)[/tex]. Here, the coefficient of [tex]\( x \)[/tex] is [tex]\(-8\)[/tex].
2. Take half of the coefficient of [tex]\( x \)[/tex]: [tex]\[ \frac{-8}{2} = -4 \][/tex]
3. Square the half value: [tex]\[ (-4)^2 = 16 \][/tex]
4. Add this square to both sides of the equation:
[tex]\[
x^2 - 8x + 16 = 9 + 16
\][/tex]
5. After adding, rewrite the equation as:
[tex]\[
x^2 - 8x + 16 = 25
\][/tex]
So, the completed square on the left-hand side is [tex]\( x^2 - 8x + 16 \)[/tex], and adding 16 to both sides results in the right-hand side being 25.
Therefore, to complete the square:
1. The value to add to both sides is [tex]\( \boxed{16} \)[/tex].
2. The left side of the equation becomes: [tex]\( x^2 - 8x + 16 \)[/tex].
3. The right side of the equation becomes: [tex]\( 25 \)[/tex].
