Question

The beginning steps for determining the center and radius of a circle using the completing the square method are shown in the table.Step 1[original equation]:x2 + 8x + y2 − 6y = 11Step 2[group like terms]:(x2 + 8x) + (y2 − 6y) = 11Step 3[complete the square]:Which of the following is the correct equation for Step 3?

Answer 1

We want to complete the square of

`(x^2+8x)+(y^2-6y)=11`

Firstly, let us look at how to complete a square.

Consider the illustration below

`\begin{gathered} To\text{ complete the square of }(x^2+8x) \\ Add\text{ \lparen}(8)/(2))^2\text{ to both sides of the equation} \\ That\text{ is 4}^2=16 \end{gathered}`
`\begin{gathered} To\text{ complete the sqaure of y}^2-6y,\text{ } \\ Add\text{ \lparen}(-6)/(2))^2=(-3)^2=9\text{ to both sides of the equation} \end{gathered}`

Thus, we have

`(x^2+8x+16)+(y^2-6y+9)=11+_16+9`
`\begin{gathered} The\text{ answer is } \\ (x^2+8x+16)+(y^2-6y+9)=11+16+9 \end{gathered}`