The slopes of the lines represented by the equation 5x^2 + 6xy + 2y^2 = 0 are:

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Question: The slopes of the lines represented by the equation 5x^2 + 6xy + 2y^2 = 0 are:

Options:

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

Solution:

The slopes can be calculated using the quadratic formula, yielding -3/5 and -2/5.

The slopes of the lines represented by the equation 5x^2 + 6xy + 2y^2 = 0 are:

The slopes of the lines represented by the equation 5x^2 + 6xy + 2y^2 = 0 are:

Step 1: Start with the given equation: 5x^2 + 6xy + 2y^2 = 0.
Step 2: Recognize that this is a quadratic equation in terms of y, which can be rearranged to the standard form: 2y^2 + 6xy + 5x^2 = 0.
Step 3: Identify the coefficients for the quadratic formula, which are: a = 2, b = 6x, and c = 5x^2.
Step 4: Use the quadratic formula to find the values of y: y = (-b ± √(b² - 4ac)) / (2a).
Step 5: Substitute the coefficients into the 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: Now the equation becomes: y = (-(6x) ± √(-4x^2)) / 4.
Step 8: Since √(-4x^2) = 2xi, the equation simplifies to: y = (-(6x) ± 2xi) / 4.
Step 9: Separate the real and imaginary parts to find the slopes: y = (-3/2)x ± (1/2)xi.
Step 10: The slopes of the lines are the coefficients of x, which are -3/5 and -2/5.

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