Question

Complete the square for x^2+y^2+y+2=8

Answer 1

Answer:
`\text{x}^2+\left(\text{y}+(1)/(2)\right)^2=(25)/(4)`

This is a circle with center (0,-1/2) and radius = 5/2 = 2.5

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Work Shown

Part 1

`\text{x}^2+\text{y}^2+\text{y}+2=8\\\\\text{x}^2+\left(\text{y}^2+\text{y}\right)+2=8\\\\\text{x}^2+\left(\text{y}^2+\text{y}+0\right)+2=8\\\\\text{x}^2+\left(\text{y}^2+\text{y}+(1)/(4)-(1)/(4)\right)+2=8\\\\`

Part 2

`\text{x}^2+\left(\text{y}^2+\text{y}+(1)/(4)\right)-(1)/(4)+2=8\\\\\text{x}^2+\left(\text{y}+(1)/(2)\right)^2+(7)/(4)=8\\\\\text{x}^2+\left(\text{y}+(1)/(2)\right)^2=8-(7)/(4)\\\\\text{x}^2+\left(\text{y}+(1)/(2)\right)^2=(25)/(4)\\\\`

When compared to the template
`(\text{x}-\text{h}\left)^2+(\text{y}-\text{k}\left)^2 = \text{r}^2\\\\` we see that the center of the circle is (h,k) = (0,-1/2) and the radius is r = sqrt(25/4) = 5/2 = 2.5

In the last step of part 1, I added 1/4 and subtracted 1/4 to help complete the square. The 1/4 comes from these steps:

• Take half of the y coefficient (1) to get 1/2
• Square 1/2 to get 1/4

GeoGebra can be used to verify the answer is correct.