To rewrite the equation [tex]\( x^2 + y^2 - 18x + 8y + 5 = 0 \)[/tex] in standard form, we need to complete the square for the [tex]\( x \)[/tex]-terms and the [tex]\( y \)[/tex]-terms. Let's go through the process step-by-step.
1. Group terms and move the constant to the other side of the equation:
[tex]\[
x^2 - 18x + y^2 + 8y = -5
\][/tex]
2. Determine the values that need to be added to both sides of the equation:
- For [tex]\( x \)[/tex]-terms:
[tex]\[
(-18 \div 2)^2 = (-9)^2 = 81
\][/tex]
- For [tex]\( y \)[/tex]-terms:
[tex]\[
(8 \div 2)^2 = 4^2 = 16
\][/tex]
3. Add the values to both sides of the equation:
[tex]\[
x^2 - 18x + 81 + y^2 + 8y + 16 = -5 + 81 + 16
\][/tex]
Simplifying the right side:
[tex]\[
x^2 - 18x + 81 + y^2 + 8y + 16 = 92
\][/tex]
4. Write each trinomial as a binomial squared, and simplify the right side:
[tex]\[
(x - 9)^2 + (y + 4)^2 = 92
\][/tex]
Therefore, the standard form of the given equation [tex]\( x^2 + y^2 - 18x + 8y + 5 = 0 \)[/tex] is:
[tex]\[
(x - 9)^2 + (y + 4)^2 = 92
\][/tex]
