The slopes of the lines represented by the equation 2x^2 + 3xy + y^2 = 0 are:

The slopes of the lines represented by the equation 2x^2 + 3xy + y^2 = 0 are:

Step 1: Start with the given equation: 2x^2 + 3xy + y^2 = 0.
Step 2: Rearrange the equation to express it in terms of y: y^2 + 3xy + 2x^2 = 0.
Step 3: Recognize that this is a quadratic equation in y, which can be written as Ay^2 + By + C = 0, where A = 1, B = 3x, and C = 2x^2.
Step 4: Use the quadratic formula to find the values of y: y = (-B ± √(B² - 4AC)) / (2A).
Step 5: Substitute A, B, and C into the quadratic formula: y = (-(3x) ± √((3x)² - 4(1)(2x^2))) / (2(1)).
Step 6: Simplify the expression under the square root: y = (-(3x) ± √(9x² - 8x²)) / 2.
Step 7: This simplifies to y = (-(3x) ± √(x²)) / 2.
Step 8: Further simplify to find y = (-(3x) ± x) / 2.
Step 9: This gives two possible values for y: y = (-4x)/2 = -2x and y = (-2x)/2 = -x.
Step 10: The slopes of the lines are the coefficients of x in these equations, which are -2 and -1.

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