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

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

Step 1: Start with the given equation of the pair of lines: 2x^2 + 3xy + y^2 = 0.
Step 2: Rewrite the equation in terms of y. This means we want to express y in terms of x and m (the slope).
Step 3: Substitute y = mx into the equation. This gives us: 2x^2 + 3x(mx) + (mx)^2 = 0.
Step 4: Simplify the equation. This results in: 2x^2 + 3mx^2 + m^2x^2 = 0.
Step 5: Factor out x^2 from the equation: x^2(2 + 3m + m^2) = 0.
Step 6: Since x^2 = 0 gives us a trivial solution, we focus on the quadratic part: 2 + 3m + m^2 = 0.
Step 7: Rearrange the equation to standard quadratic form: m^2 + 3m + 2 = 0.
Step 8: Factor the quadratic equation: (m + 1)(m + 2) = 0.
Step 9: Set each factor to zero to find the slopes: m + 1 = 0 gives m = -1, and m + 2 = 0 gives m = -2.
Step 10: Conclude that the slopes of the lines are -1 and -2.

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