Find the slopes of the lines represented by the equation 5x^2 + 6xy + 2y^2 = 0.

Find the slopes of the lines represented by the equation 5x^2 + 6xy + 2y^2 = 0.

Correct Answer: m = -3, m = -2/5

Step 1: Start with the given equation: 5x^2 + 6xy + 2y^2 = 0.
Step 2: Rearrange the equation to express it in terms of y: 2y^2 + 6xy + 5x^2 = 0.
Step 3: Identify this as a quadratic equation in y: Ay^2 + By + C = 0, where A = 2, B = 6x, and C = 5x^2.
Step 4: Use the quadratic formula to find y: y = (-B ± √(B² - 4AC)) / (2A).
Step 5: Substitute A, B, and C into the quadratic formula: y = (-(6x) ± √((6x)² - 4(2)(5x^2))) / (2(2)).
Step 6: Simplify the expression under the square root: (6x)² - 4(2)(5x^2) = 36x^2 - 40x^2 = -4x^2.
Step 7: Substitute this back into the formula: y = (-(6x) ± √(-4x^2)) / 4.
Step 8: Recognize that √(-4x^2) = 2xi, where i is the imaginary unit.
Step 9: Thus, y = (-(6x) ± 2xi) / 4 = (-3/2)x ± (1/2)xi.
Step 10: The slopes of the lines can be found from the expression for y in terms of x, which gives us the slopes as m = (-3/2) ± (1/2)i.

Implicit Differentiation – The question involves finding the slopes of curves defined by implicit equations, requiring the use of implicit differentiation to derive the relationship between x, y, and their derivatives.
Quadratic Equations – The equation is quadratic in form, and understanding how to manipulate and solve quadratic equations is essential to find the slopes.