Final answer:
The equation 4x²-16x+4y²+24y+36=0 is transformed into standard form by completing the square which results in the circle equation (x-2)² + (y+3)² = 4.
Step-by-step explanation:
The equation 4x²-16x+4y²+24y+36=0 is a quadratic equation in two variables x and y. to transform this equation into standard form, we complete the square for both x and y terms.
First, group the x terms and the y terms:
(4x²-16x) + (4y²+24y) + 36 = 0
Next, factor out the coefficients of the quadratic terms from each group:
4(x²-4x) + 4(y²+6y) + 36 = 0
Now, complete the square for each group:
- Add and subtract (4/2)² inside the first parenthesis: 4(x²-4x+4-4)
- Add and subtract (6/2)² inside the second parenthesis: 4(y²+6y+9-9)
Now the equation becomes:
4((x-2)² - 4) + 4((y+3)² - 9) + 36 = 0
Simplify by distributing the 4 and combining like terms:
4(x-2)² - 16 + 4(y+3)² - 36 + 36 = 0
This simplifies further to:
4(x-2)² + 4(y+3)² = 16
Finally, divide throughout by 16 to get the equation in standard form:
(x-2)² + (y+3)² = 4
This represents a circle with a radius of 2 centered at (2, -3).