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There are four steps for converting the equation [tex]\( x^2 + y^2 + 12x + 2y - 1 = 0 \)[/tex] into standard form by completing the square. Complete the last step.

1. Group the [tex]\( x \)[/tex] terms together and the [tex]\( y \)[/tex] terms together, and move the constant term to the other side of the equation.
[tex]\[ x^2 + 12x + y^2 + 2y = 1 \][/tex]

2. Determine [tex]\(\left(\frac{b}{2}\right)^2\)[/tex] for the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] terms.
[tex]\[ \left(\frac{12}{2}\right)^2 = 36 \quad \text{and} \quad \left(\frac{2}{2}\right)^2 = 1 \][/tex]

3. Add the values to both sides of the equation.
[tex]\[ x^2 + 12x + 36 + y^2 + 2y + 1 = 1 + 36 + 1 \][/tex]

4. Write each trinomial as a binomial squared, and simplify the right side.
[tex]\[ (x + \square)^2 + (y + \square)^2 = \square \][/tex]



Answer :

GinnyAnswer
To complete the last step:

First, we write each trinomial as a binomial squared:

- The [tex]\(x\)[/tex]-terms: [tex]\(x^2 + 12x + 36\)[/tex]
[tex]\[ x^2 + 12x + 36 = (x + 6)^2 \][/tex]

- The [tex]\(y\)[/tex]-terms: [tex]\(y^2 + 2y + 1\)[/tex]
[tex]\[ y^2 + 2y + 1 = (y + 1)^2 \][/tex]

Next, we simplify the right side of the equation:

- Combining all constants on the right-hand side:
[tex]\[ 1 + 36 + 1 = 38 \][/tex]

Therefore, the equation becomes:
[tex]\[ (x + 6)^2 + (y + 1)^2 = 38 \][/tex]

So, the final result in standard form is:

[tex]\[ (x + 6)^2 + (y + 1)^2 = 38 \][/tex]