To solve the equation [tex]\( x^2 - 8x = 9 \)[/tex] by completing the square, follow these steps:
1. Identify the coefficient of [tex]\(x\)[/tex]:
In the given equation [tex]\( x^2 - 8x = 9 \)[/tex], the coefficient of [tex]\(x\)[/tex] is [tex]\(-8\)[/tex].
2. Calculate half of the coefficient of [tex]\(x\)[/tex] and square it:
Take the coefficient of [tex]\(x\)[/tex], which is [tex]\(-8\)[/tex], and divide it by 2 to get [tex]\(-4\)[/tex]. Then square this result:
[tex]\[
\left( \frac{-8}{2} \right)^2 = (-4)^2 = 16
\][/tex]
3. Add this square to both sides of the equation:
We need to add [tex]\(16\)[/tex] to both sides of the equation to complete the square:
[tex]\[
x^2 - 8x + 16 = 9 + 16
\][/tex]
4. Write the new equation with the completed square:
Now the equation becomes:
[tex]\[
x^2 - 8x + 16 = 25
\][/tex]
5. Express the left-hand side as a perfect square:
The left-hand side of the equation [tex]\( x^2 - 8x + 16 \)[/tex] is a perfect square trinomial and can be rewritten as:
[tex]\[
(x - 4)^2 = 25
\][/tex]
So, by following these steps, we completed the square by adding [tex]\(16\)[/tex] to both sides, yielding the equation [tex]\( (x - 4)^2 = 25 \)[/tex].
