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 $(b/2)^2$ for the $x$ and $y$ terms.
$(12/2)^2=36 \text { and }(2/2)^2=1$
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

Complete the square for the x terms: x 2 + 12 x + 36 = ( x + 6 ) 2 .
Complete the square for the y terms: y 2 + 2 y + 1 = ( y + 1 ) 2 .
Simplify the right side of the equation: 1 + 36 + 1 = 38 .
Write the final equation in standard form: ( x + 6 ) 2 + ( y + 1 ) 2 = 38 , so the missing values are 6 , 1 , 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 to convert it into standard form by completing the square. Our goal is to complete the last step, which involves writing each trinomial as a binomial squared and simplifying the right side of the equation.

Rewriting the Equation From step 3, we have x 2 + 12 x + 36 + y 2 + 2 y + 1 = 1 + 36 + 1 . We need to rewrite the left side as ( x + □ ) 2 + ( y + □ ) 2 and simplify the right side.

Completing the Square for x For the x terms, we have x 2 + 12 x + 36 . This is a perfect square trinomial and can be written as ( x + 6 ) 2 since ( x + 6 ) 2 = x 2 + 2 ( 6 ) x + 6 2 = x 2 + 12 x + 36 .

Completing the Square for y For the y terms, we have y 2 + 2 y + 1 . This is also a perfect square trinomial and can be written as ( y + 1 ) 2 since ( y + 1 ) 2 = y 2 + 2 ( 1 ) y + 1 2 = y 2 + 2 y + 1 .

Simplifying the Right Side Now, let's simplify the right side of the equation: 1 + 36 + 1 = 38 .

Final Equation Therefore, the equation becomes ( x + 6 ) 2 + ( y + 1 ) 2 = 38 . The missing values are 6, 1, and 38.


Examples
Completing the square is a useful technique in various fields, such as physics and engineering, for solving problems involving quadratic equations. For example, in projectile motion, the equation describing the height of a projectile as a function of time is a quadratic equation. By completing the square, we can easily find the maximum height reached by the projectile and the time at which it occurs. Suppose the height of a ball thrown upwards is given by h ( t ) = − 5 t 2 + 30 t + 2 . Completing the square allows us to rewrite this as h ( t ) = − 5 ( t − 3 ) 2 + 47 , revealing that the maximum height of 47 meters is reached at time t=3 seconds.

Answered by GinnyAnswer | 2025-07-08

The final equation in standard form after completing the squares is ( x + 6 ) 2 + ( y + 1 ) 2 = 38 . The values for completion are 6, 1, and 38. This indicates the centers of the circle and its radius squared.
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Answered by Anonymous | 2025-07-17