Find the slopes of the lines represented by the equation 6x^2 - 5xy + y^2 = 0.

Find the slopes of the lines represented by the equation 6x^2 - 5xy + y^2 = 0.

Step 1: Start with the given equation: 6x^2 - 5xy + y^2 = 0.
Step 2: Rearrange the equation to express it in a standard quadratic form in terms of y: y^2 - 5xy + 6x^2 = 0.
Step 3: Identify the coefficients in the quadratic equation: A = 1 (coefficient of y^2), B = -5x (coefficient of y), C = 6x^2 (constant term).
Step 4: Use the quadratic formula to find the values of y: y = (-B ± √(B^2 - 4AC)) / (2A).
Step 5: Substitute A, B, and C into the quadratic formula: y = (5x ± √((-5x)^2 - 4(1)(6x^2))) / (2(1)).
Step 6: Simplify the expression under the square root: y = (5x ± √(25x^2 - 24x^2)) / 2 = (5x ± √(x^2)) / 2.
Step 7: Further simplify: y = (5x ± x) / 2.
Step 8: Split into two cases: y = (5x + x) / 2 = 3x and y = (5x - x) / 2 = 2x.
Step 9: Identify the slopes from the equations: The slope of y = 3x is 3, and the slope of y = 2x is 2.
Step 10: The slopes can also be expressed as fractions: The slopes are 5/6 and -1.

Quadratic Equations – Understanding how to derive slopes from a quadratic equation in two variables.
Implicit Differentiation – Applying implicit differentiation to find the slopes of curves defined by equations.
Roots of Equations – Finding the roots of a quadratic equation to determine the slopes of the tangent lines.