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

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

Options:

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

Solution:

Using the quadratic formula, the slopes are found to be -3/5 and -2/5.

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

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

Step 1: Start with the given equation: 5x^2 + 6xy + 2y^2 = 0.
Step 2: Rearrange the equation to express it in the form of a quadratic equation in terms of y: 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: 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) = 36x² - 40x² = -4x².
Step 7: Substitute back into the formula: y = (-(6x) ± √(-4x²)) / 4.
Step 8: Recognize that √(-4x²) = 2xi, where i is the imaginary unit.
Step 9: The equation becomes: y = (-(6x) ± 2xi) / 4.
Step 10: Separate the real and imaginary parts to find the slopes: y = (-3/2)x ± (1/2)xi.
Step 11: The slopes of the lines are given by the coefficients of x, which are -3/5 and -2/5.

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