Part A: Complete the square to rewrite the following equation in standard form. Show all necessary work. (6 points)

Part B: What are the center and radius of the circle? (4 points)

Ask by Brewer Flynn. in the United States

Mar 26,2025

Question

Part A: Complete the square to rewrite the following equation in standard form. Show all necessary work. (6 points)

Part B: What are the center and radius of the circle? (4 points)

Ask by Brewer Flynn. in the United States

Mar 26,2025

UpStudy AI · Accepted Answer

**Step 1: Group and Rearrange Terms**

We start with the equation  
$$
x^{2} - 4x + y^{2} + 8y = -4.
$$

Group the $$x$$-terms and $$y$$-terms together:  
$$
(x^{2} - 4x) + (y^{2} + 8y) = -4.
$$

**Step 2: Complete the Square for the $$x$$-terms**

For $$x^2-4x$$, take half of the coefficient of $$x$$ (which is $$-4$$):  
$$
\frac{-4}{2} = -2.
$$  
Square $$-2$$ to get:  
$$
(-2)^2 = 4.
$$  
Add and subtract $$4$$ inside the parentheses:  
$$
x^{2} - 4x = (x^{2} - 4x + 4) - 4 = (x-2)^2 - 4.
$$

**Step 3: Complete the Square for the $$y$$-terms**

For $$y^2+8y$$, take half of the coefficient of $$y$$ (which is $$8$$):  
$$
\frac{8}{2} = 4.
$$  
Square $$4$$ to get:  
$$
4^2 = 16.
$$  
Add and subtract $$16$$ in the grouping:  
$$
y^{2} + 8y = (y^{2} + 8y + 16) - 16 = (y+4)^2 - 16.
$$

**Step 4: Rewrite the Equation in Standard Form**

Substitute the completed squares back into the equation:
$$
(x-2)^2 - 4 + (y+4)^2 - 16 = -4.
$$
Combine the constant terms:
$$
(x-2)^2 + (y+4)^2 - 20 = -4.
$$
Add $$20$$ to both sides to isolate the squared terms:
$$
(x-2)^2 + (y+4)^2 = 16.
$$

**Step 5: Identify the Center and Radius**

The standard form of a circle is:
$$
(x-h)^2 + (y-k)^2 = r^2,
$$
where $$(h,k)$$ is the center and $$r$$ is the radius.

By comparing:
$$
(x-2)^2 + (y+4)^2 = 16,
$$
we have:
- Center: $$(2, -4)$$
- Radius: $$ r = \sqrt{16} = 4 $$.
**Part A: Standard Form of the Circle** $$ (x-2)^2 + (y+4)^2 = 16 $$ **Part B: Center and Radius** - **Center:** $$(2, -4)$$ - **Radius:** $$4$$

Part A: Complete the square to rewrite the following equation in standard form. Show all necessary work. (6 points)

Part B: What are the center and radius of the circle? (4 points)

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Geometry Questions

Part A: Complete the square to rewrite the following equation in standard form. Show all necessary work. (6 points) Part B: What are the center and radius of the circle? (4 points)

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Geometry Questions

Part A: Complete the square to rewrite the following equation in standard form. Show all necessary work. (6 points)

Part B: What are the center and radius of the circle? (4 points)