Final Answer:
The general form of the parabola with the equation (y-3)²=6(x-8) is:
y² = 6x + 6y - 57.
Step-by-step explanation:
To transform the equation (y-3)² = 6(x-8) into the general form of a parabola, expand the squared term:
(y-3)² = 6(x-8)
y² - 6y + 9 = 6x - 48
Rearrange terms to put it in the general form of a parabola:
y² - 6y = 6x - 48 - 9
y² - 6y = 6x - 57
Now, complete the square for the y terms by adding and subtracting the square of half the coefficient of y (which is (-6/2) = -3):
y² - 6y + (-3)² = 6x - 57 + (-3)²
y² - 6y + 9 = 6x - 57 + 9
y² - 6y + 9 = 6x - 48
Factor the left-hand side as a perfect square:
(y - 3)² = 6x - 48
To get the general form y² = ax + bx + c, expand the squared term:
(y - 3)² = 6x - 48
y² - 6y + 9 = 6x - 48
Rearrange terms to obtain the general form:
y² = 6x - 48 - 9 + 6y
y² = 6x - 57 + 6y
y² = 6x + 6y - 57
Finally, arrange the terms in descending order:
y² = 6x - 57 + 6y
y² = 6x + 6y - 57
Therefore, the general form of the parabola's equation is y² = 6x + 6y - 57.