To complete the square for the equation [tex]\( x^2 - 8x = 9 \)[/tex], follow these steps:
1. Rewrite the equation in standard form:
[tex]\[ x^2 - 8x = 9 \][/tex]
2. Identify the coefficient of [tex]\( x \)[/tex]:
The coefficient of [tex]\( x \)[/tex] is [tex]\(-8\)[/tex].
3. Find the number to add to each side to complete the square:
Take half of the coefficient of [tex]\( x \)[/tex], and then square it. This number is:
[tex]\[
\left(\frac{-8}{2}\right)^2 = (-4)^2 = 16
\][/tex]
4. Add 16 to both sides of the equation:
[tex]\[
x^2 - 8x + 16 = 9 + 16
\][/tex]
5. Simplify the right side of the equation:
[tex]\[
x^2 - 8x + 16 = 25
\][/tex]
6. Write the left side as a squared binomial:
[tex]\[
(x - 4)^2 = 25
\][/tex]
So, in summary, you add 16 to both sides to complete the square. The completed square form of the equation [tex]\( x^2 - 8x = 9 \)[/tex] is [tex]\( (x - 4)^2 = 25 \)[/tex].
To find the solutions to the equation, you can solve for [tex]\( x \)[/tex] from the completed square form:
7. Take the square root of both sides:
[tex]\[
\sqrt{(x - 4)^2} = \pm \sqrt{25}
\][/tex]
This simplifies to:
[tex]\[
x - 4 = \pm 5
\][/tex]
8. Solve for [tex]\( x \)[/tex]:
[tex]\[
x - 4 = 5 \quad \text{or} \quad x - 4 = -5
\][/tex]
Therefore:
[tex]\[
x = 9 \quad \text{or} \quad x = -1
\][/tex]
The solutions to the original equation [tex]\( x^2 - 8x = 9 \)[/tex] are [tex]\( x = 9 \)[/tex] and [tex]\( x = -1 \)[/tex].
