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

Verified step by step guidance

Start with the given equation: $4x^2 - 2xy + 3y^2 = 2$.

Rewrite the equation to isolate terms involving $y$: $3y^2 - 2xy + 4x^2 = 2$.

Recognize this as a quadratic equation in terms of $y$, where $a = 3$, $b = -2x$, and $c = 4x^2 - 2$.

Use the quadratic formula to solve for $y$: $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, substituting $a$, $b$, and $c$ accordingly.

Simplify the expression under the square root and write the solution for $y$ explicitly in terms of $x$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Solving for a Variable

Solving for a variable means isolating that variable on one side of the equation. In this problem, you need to express y explicitly in terms of x, which may involve rearranging terms and using algebraic techniques to isolate y.

Quadratic Equations in Two Variables

The given equation is quadratic in y because it contains y squared and a product term xy. Understanding how to handle quadratic equations with two variables is essential, as you may need to use methods like the quadratic formula to solve for y.

Using the Quadratic Formula

When an equation is quadratic in y, the quadratic formula can be used to solve for y in terms of x. The formula y = [-b ± sqrt(b² - 4ac)] / 2a helps find the roots of the quadratic equation, where a, b, and c are expressions involving x.

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