To convert the given equation $x^2 + y^2 - 6y - 2x = 15$ into standard form, which is the form of a circle, we need to complete the square for both the $x$-terms and the $y$-terms.

Step 1: Group the $x$-terms and $y$-terms

$x^2 - 2x + y^2 - 6y = 15$

Step 2: Complete the square for $x$-terms

• For $x^2 - 2x$, take half of $-2$ (which is $-1$), and then square it: $(-1)^2 = 1$.
• Add 1 to both sides of the equation.

Step 3: Complete the square for $y$-terms

• For $y^2 - 6y$, take half of $-6$ (which is $-3$), and then square it: $(-3)^2 = 9$.
• Add 9 to both sides of the equation.

Step 4: Rewrite the equation

Now, rewrite the equation with the squares completed: $(x^2 - 2x + 1) + (y^2 - 6y + 9) = 15 + 1 + 9$ This simplifies to: $(x - 1)^2 + (y - 3)^2 = 25$

Step 5: The standard form of the equation

The equation is now in the standard form of a circle: $(x - 1)^2 + (y - 3)^2 = 25$ This represents a circle with:

• Center: $(1, 3)$
• Radius: $5$ (since $\sqrt{25} = 5$).

Would you like more details on completing the square, or have any other questions?

Here are 5 related questions you can explore next:

1. How do you complete the square for an equation like $x^2 + 4x + 7 = 0$?
2. What is the general form of the equation of a circle, and how can you identify its center and radius?
3. How do you convert the equation of a circle into general form from standard form?
4. Can you determine the intersection points of two circles given their equations?
5. What is the geometric meaning of completing the square in the context of conic sections?

Tip: Completing the square often helps when converting quadratic equations into forms that are easier to graph or analyze geometrically, especially for circles, parabolas, ellipses, and hyperbolas.