x²+y²-16x-8y-20=0 express to standard form

To express the equation \( x^2 + y^2 - 16x - 8y - 20 = 0 \) in standard form, we will complete the square for both \( x \) and \( y \).

### Step 1: Rearrange the terms
Group the \( x \)-terms and \( y \)-terms together:

\[
(x^2 - 16x) + (y^2 - 8y) = 20
\]

### Step 2: Complete the square for \( x \)
Take the coefficient of \( x \), which is \( -16 \), divide it by 2 to get \( -8 \), and then square it to get \( 64 \). Add and subtract \( 64 \):

\[
(x^2 - 16x) = (x - 8)^2 - 64
\]

### Step 3: Complete the square for \( y \)
Take the coefficient of \( y \), which is \( -8 \), divide it by 2 to get \( -4 \), and then square it to get \( 16 \). Add and subtract \( 16 \):

\[
(y^2 - 8y) = (y - 4)^2 - 16
\]

### Step 4: Substitute back into the equation
Now, substitute the completed squares into the original equation:

\[
((x - 8)^2 - 64) + ((y - 4)^2 - 16) = 20
\]

### Step 5: Simplify
Combine the constants:

\[
(x - 8)^2 + (y - 4)^2 - 80 = 20
\]

Add 80 to both sides:

\[
(x - 8)^2 + (y - 4)^2 = 100
\]

### Final Answer: Standard Form
The equation in standard form is:

\[
(x - 8)^2 + (y - 4)^2 = 100
\]

This represents a circle with center \( (8, 4) \) and radius \( \sqrt{100} = 10 \).

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Would you like more details on any of the steps? 

Here are 5 related questions:
1. How do you complete the square for more complex expressions?
2. What is the geometric interpretation of this standard form equation?
3. How would this process change for an ellipse equation?
4. Can you explain the general formula for the equation of a circle?
5. How do you solve equations involving circles intersecting with lines?

**Tip:** Completing the square is essential for converting quadratic equations into more understandable forms, like circles and parabolas.

x² + y² - 16x - 8y - 20 = 0 express to standard form

Learn how to convert the equation x² + y² - 16x - 8y - 20 = 0 into the standard form of a circle. This solution explains the step-by-step process of completing the square and provides the final form of the equation, highlighting the circle's center at (8, 4) and radius 10. Ideal for students in Grades 9-10.

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