Alright, let's solve the equation [tex]\( x^2 - 8x = 9 \)[/tex] by completing the square step by step.
1. Identify the coefficient of [tex]\( x \)[/tex] in the given equation.
- The given equation is [tex]\( x^2 - 8x = 9 \)[/tex].
- The coefficient of [tex]\( x \)[/tex] is [tex]\(-8\)[/tex].
2. Calculate the value to be added to both sides to complete the square.
- To complete the square, we take half of the coefficient of [tex]\( x \)[/tex], which is [tex]\(-8\)[/tex], then square it.
- Half of [tex]\(-8\)[/tex] is [tex]\(-4\)[/tex]. Squaring [tex]\(-4\)[/tex] gives us [tex]\( (-4)^2 = 16 \)[/tex].
3. Add this value to both sides of the equation.
- Adding 16 to the left side: [tex]\( x^2 - 8x + 16 \)[/tex].
- Adding 16 to the right side: [tex]\( 9 + 16 \)[/tex].
4. Rewrite the left side as a squared binomial.
- The left side [tex]\( x^2 - 8x + 16 \)[/tex] can be written as [tex]\( (x - 4)^2 \)[/tex].
Therefore, the equation after completing the square looks like this:
[tex]\[ (x - 4)^2 = 25 \][/tex]
In summary:
- The value added to both sides of the equation is [tex]\( 16 \)[/tex].
- The left side after adding this value and completing the square is [tex]\( x^2 - 8x + 16 \)[/tex].
- The right side after adding this value is [tex]\( 25 \)[/tex].
Thus, we have:
[tex]\[ x^2 - 8x + 16 = 25 \][/tex]
This is the completed square form of the given equation.
