Final answer:
To find the equation of the tangent line to the curve at the given point, we can use implicit differentiation and the point-slope form of a line.
Step-by-step explanation:
To find the equation of the tangent line to the curve at the point (0, -2), we can use implicit differentiation. Let's start by differentiating both sides of the equation y²(y² - 4) = x²(x² - 6) with respect to x.
On the left-hand side, we can apply the product rule and chain rule, which gives 2yy'(y² - 4) + y²(2yy') = 2x(x² - 6) + x²(2x).
Next, we substitute the coordinates of the given point, (0, -2), into the equation and solve for y'. Plugging in x = 0 and y = -2, we have -8y' + 4 = 0, which gives y' = 1/2.
Therefore, the equation of the tangent line to the curve at the point (0, -2) is y = (1/2)x - 2.