There are four steps for converting the equation $x^2+y^2+12 x+2 y-1=0$ into standard form by completing the square. Complete the last step.
1. Group the $x$ terms together and the $y$ terms together, and move the constant term to the other side of the equation.
$x^2+12 x+y^2+2 y=1$
2. Determine $(b \div 2)^2$ for the $x$ and $y$ terms.
$(12 \div 2)^2=36 \text { and }(2 \div 2)^2=1$
3. Add the values to both sides of the equation.
$x^2+12 x+36+y^2+2 y+1=1+36+1$
4. Write each trinomial as a binomial squared, and simplify the right side.
$(x+\square)^2+(y+\square)^2=\square
Answer (1)
Group the x and y terms and move the constant: x 2 + 12 x + y 2 + 2 y = 1 .
Calculate the values to complete the square: ( 12/2 ) 2 = 36 and ( 2/2 ) 2 = 1 .
Add these values to both sides: x 2 + 12 x + 36 + y 2 + 2 y + 1 = 1 + 36 + 1 .
Write the equation in standard form: ( x + 6 ) 2 + ( y + 1 ) 2 = 38 . The final answer is ( x + 6 ) 2 + ( y + 1 ) 2 = 38 .
Explanation
Understanding the Problem We are given the equation x 2 + y 2 + 12 x + 2 y − 1 = 0 and the first three steps of completing the square to convert it into standard form. Our goal is to complete the last step and write the equation in the standard form of a circle.
Step 1: Grouping Terms The first step is to group the x and y terms and move the constant to the right side, resulting in x 2 + 12 x + y 2 + 2 y = 1 .
Step 2: Completing the Square Values The second step is to find the values to complete the square for both x and y . For x , we have ( 12/2 ) 2 = 6 2 = 36 , and for y , we have ( 2/2 ) 2 = 1 2 = 1 .
Step 3: Adding to Both Sides The third step is to add these values to both sides of the equation: x 2 + 12 x + 36 + y 2 + 2 y + 1 = 1 + 36 + 1 .
Step 4: Writing in Standard Form Now, we complete the square for the x and y terms. We have x 2 + 12 x + 36 = ( x + 6 ) 2 and y 2 + 2 y + 1 = ( y + 1 ) 2 . Also, we simplify the right side: 1 + 36 + 1 = 38 . Therefore, the equation becomes ( x + 6 ) 2 + ( y + 1 ) 2 = 38 .
Final Answer Finally, we fill in the blanks in the given expression: ( x + 6 ) 2 + ( y + 1 ) 2 = 38 .
Examples
Completing the square is a useful technique in various fields. For example, in physics, it can be used to find the center of mass of an object or to solve differential equations. In engineering, it can be used to design optimal control systems. In everyday life, understanding circles and their equations can help in designing circular gardens or understanding the coverage area of a Wi-Fi router.
