Sure, let's put the equation [tex]\( x^2 + y^2 + 8x - 16y + 4 = 0 \)[/tex] into standard form. Follow these steps:
1. Complete the square for the [tex]\(x\)[/tex]-terms:
- Start with [tex]\( x^2 + 8x \)[/tex].
- To complete the square, take half of the coefficient of [tex]\(x\)[/tex] (which is 8), and square it. [tex]\[\left( \frac{8}{2} \right)^2 = 16.\][/tex]
- Add and subtract this value within the equation:
[tex]\[ x^2 + 8x = (x + 4)^2 - 16. \][/tex]
2. Complete the square for the [tex]\(y\)[/tex]-terms:
- Start with [tex]\( y^2 - 16y \)[/tex].
- To complete the square, take half of the coefficient of [tex]\(y\)[/tex] (which is -16), and square it. [tex]\[\left( \frac{-16}{2} \right)^2 = 64.\][/tex]
- Add and subtract this value within the equation:
[tex]\[ y^2 - 16y = (y - 8)^2 - 64. \][/tex]
3. Rewrite the entire equation with these completed squares:
[tex]\[ x^2 + 8x + y^2 - 16y + 4 = \left( (x + 4)^2 - 16 \right) + \left( (y - 8)^2 - 64 \right) + 4 = 0. \][/tex]
4. Simplify the equation:
[tex]\[ (x + 4)^2 - 16 + (y - 8)^2 - 64 + 4 = 0. \][/tex]
[tex]\[ (x + 4)^2 + (y - 8)^2 - 76 = 0. \][/tex]
5. Move the constant term to the other side of the equation:
[tex]\[ (x + 4)^2 + (y - 8)^2 = 76. \][/tex]
Thus, the standard form of the given equation is:
[tex]\[ (x + 4)^2 + (y - 8)^2 = 76. \][/tex]
In this form:
[tex]\[ (x + [?])^2 = (x + 4)^2 \rightarrow \text{?} = 4, \][/tex]
[tex]\[ (y - \square)^2 = (y - 8)^2 \rightarrow \square = 8. \][/tex]
