Find the equation of the pair of lines represented by the equation 2x^2 + 3xy +

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Question: Find the equation of the pair of lines represented by the equation 2x^2 + 3xy + y^2 = 0.

Options:

Correct Answer: y = -2x, y = -x/3

Solution:

Using the quadratic formula for the slopes gives m1 = -2 and m2 = -1/3.

Find the equation of the pair of lines represented by the equation 2x^2 + 3xy + y^2 = 0.

Find the equation of the pair of lines represented by the equation 2x^2 + 3xy + y^2 = 0.

Correct Answer: y = -2x and y = -1/3 x

Step 1: Start with the given equation: 2x^2 + 3xy + y^2 = 0.
Step 2: Rewrite the equation in the form of a quadratic in y: y^2 + 3xy + 2x^2 = 0.
Step 3: Identify the coefficients for the quadratic formula: a = 1, b = 3x, c = 2x^2.
Step 4: Use the quadratic formula to find the roots (slopes) of the equation: y = (-b ± √(b² - 4ac)) / 2a.
Step 5: Substitute a, b, and c into the quadratic formula: y = (-(3x) ± √((3x)² - 4(1)(2x²))) / (2(1)).
Step 6: Simplify the expression under the square root: (3x)² - 8x² = -5x².
Step 7: Calculate the roots: y = (-3x ± √(-5x²)) / 2.
Step 8: Since we have a negative under the square root, we can express the slopes as m1 = -2 and m2 = -1/3.
Step 9: The pair of lines can be represented as y = m1*x and y = m2*x.

Quadratic Equations – Understanding how to manipulate and solve quadratic equations to find the slopes of lines.
Pair of Lines – Recognizing that a quadratic equation in two variables can represent a pair of straight lines.
Slope Calculation – Using the quadratic formula to find the slopes of the lines represented by the equation.