Transform the equation to standard form
3x²+3y²-12x-6y-30=0
See your posting (as Zach) below.
transform the equation y=3×2-12×-1 into its standard form
To transform the equation to standard form, we need to rearrange the terms to isolate the variables on one side and move the constant term to the other side. Follow these step-by-step instructions:
1. Start with the given equation: 3x² + 3y² - 12x - 6y - 30 = 0.
2. Group the x-terms together and the y-terms together: (3x² - 12x) + (3y² - 6y) - 30 = 0.
3. Factor out the common coefficient for each group: 3(x² - 4x) + 3(y² - 2y) - 30 = 0.
4. Complete the square for the x-terms and the y-terms separately.
For the x-terms:
a. Take half of the coefficient of x, which is -4, and square it: (-4/2)² = 4.
b. Add this value inside the parentheses and subtract it outside the parentheses: 3(x² - 4x + 4) - 3(4) + 3(y² - 2y) - 30 = 0.
For the y-terms:
a. Take half of the coefficient of y, which is -2, and square it: (-2/2)² = 1.
b. Add this value inside the parentheses and subtract it outside the parentheses: 3(x² - 4x + 4) - 3(4) + 3(y² - 2y + 1) - 3(1) - 30 = 0.
5. Simplify each parentheses expression: 3(x - 2)² - 12 + 3(y - 1)² - 3 - 30 = 0.
6. Combine like terms: 3(x - 2)² + 3(y - 1)² - 45 = 0.
7. Move the constant term to the other side: 3(x - 2)² + 3(y - 1)² = 45.
Therefore, the equation is transformed into standard form as: 3(x - 2)² + 3(y - 1)² = 45.