For each equation, (a) solve for x in terms of y. See Example 8. 2x^2 + 4xy - 3y^2 = 2

Verified step by step guidance

1. Start by identifying the given equation: \(2x^2 + 4xy - 3y^2 = 2\).
2. Rearrange the equation to isolate terms involving \(x\) on one side: \(2x^2 + 4xy = 3y^2 + 2\).
3. Factor out \(x\) from the left side: \(x(2x + 4y) = 3y^2 + 2\).
4. Divide both sides by \(2x + 4y\) to solve for \(x\): \(x = \frac{3y^2 + 2}{2x + 4y}\).
5. Simplify the expression if possible, keeping \(x\) in terms of \(y\).

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0. In the context of the given equation, it can be rearranged to fit this form, allowing us to apply methods such as factoring, completing the square, or using the quadratic formula to solve for x in terms of y.

Solving for a Variable

Solving for a variable involves isolating that variable on one side of the equation. In this case, we need to manipulate the equation to express x solely in terms of y. This often requires algebraic techniques such as rearranging terms, factoring, or applying the quadratic formula to find the values of x that correspond to a given y.

The Quadratic Formula

The quadratic formula, x = (-b ± √(b^2 - 4ac)) / (2a), provides a method for finding the roots of a quadratic equation. It is particularly useful when the equation cannot be easily factored. In this problem, once the equation is arranged in standard form, the quadratic formula can be applied to derive the solutions for x in terms of y.