For the lines represented by the equation 2x^2 + 3xy + y^2 = 0, find the slopes

For the lines represented by the equation 2x^2 + 3xy + y^2 = 0, find the slopes of the lines.

For the lines represented by the equation 2x^2 + 3xy + y^2 = 0, find the slopes of the lines.

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: This is a quadratic equation in y. Identify the coefficients: a = 1, b = 3x, c = 2x^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 = (-(3x) ± √((3x)² - 4(1)(2x²))) / (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 cases: y = (-3x + x) / 2 and y = (-3x - x) / 2.
Step 10: Simplify both cases to find the slopes: y = -x and y = -2x.
Step 11: The slopes of the lines are -1 and -2.

Quadratic Equations – Understanding how to manipulate and solve quadratic equations to find slopes of lines.
Factoring and Roots – Applying factoring techniques or the quadratic formula to find the roots, which correspond to the slopes.
Conic Sections – Recognizing that the given equation represents a pair of lines, which can be derived from a conic section.