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

Verified step by step guidance

1. Start with the given equation: \(2x^2 + 4xy - 3y^2 = 2\).
2. Rearrange the equation to isolate terms involving \(y\) on one side: \(4xy - 3y^2 = 2 - 2x^2\).
3. Factor out \(y\) from the left side: \(y(4x - 3y) = 2 - 2x^2\).
4. To solve for \(y\), divide both sides by \(4x - 3y\): \(y = \frac{2 - 2x^2}{4x - 3y}\).
5. This equation can be further simplified or solved using specific values of \(x\) or additional methods if needed.

Key Concepts

Solving Quadratic Equations

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0. To solve for y in terms of x, we often rearrange the equation into standard form and apply methods such as factoring, completing the square, or using the quadratic formula. Understanding how to manipulate these equations is crucial for isolating variables.

Isolating Variables

Isolating a variable involves rearranging an equation to express one variable in terms of another. This process often requires using algebraic operations such as addition, subtraction, multiplication, and division. Mastery of this concept is essential for solving equations where one variable needs to be expressed in terms of another, as in the given problem.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler components, or factors, that can be multiplied together to yield the original polynomial. This technique is particularly useful in solving quadratic equations, as it can simplify the process of finding the roots or solutions of the equation. Understanding how to factor is key to solving for y in the given equation.