Provide an appropiate response. At the two points where the curve x² + 2xy + y² = 9 crosses the x-axis, the tangents to the curve are parallel. What is the common slope of these tangents?

-1

Step-by-step explanation:

We can find the tangents to the curve at x = 0, where the roots of the function occur (function crosses the x-axis). We can differentiate the function with respect to x by using implicit differentiation.

We will need to use the Power Rule and Product Rule to differentiate this function.

Distribute 2 inside the parentheses.

Move all terms containing dy/dx to the left side of the equation, and all other terms to the right side of the equation.

Factor dy/dx from the left side of the equation.

Divide both sides of the equation by 2x + 2y.

Factor the negative sign out of the numerator.

Cancel out the numerator and the denominator.

The derivative of the function is -1 at all points, meaning that everywhere the function has a slope, or tangent, of -1.

The common slope of the tangents as the curve x² + 2xy + y² = 9 crosses the x-axis is -1.