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

Verified step by step guidance

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

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

Recognize this as a quadratic equation in terms of $y$: $-3y^{2} + 4xy - (2 - 2x^{2}) = 0$.

Use the quadratic formula to solve for $y$, where $a = -3$, $b = 4x$, and $c = -(2 - 2x^{2})$. The formula is $y = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$.

Substitute the values of $a$, $b$, and $c$ into the quadratic formula and simplify under the square root to express $y$ 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 Quadratic Equations in Terms of One Variable

This involves isolating one variable (y) in an equation that may be quadratic in that variable. The goal is to express y explicitly as a function of x, often requiring rearrangement and use of the quadratic formula when y appears squared.

Quadratic Formula

The quadratic formula, y = [-b ± √(b² - 4ac)] / (2a), is used to solve quadratic equations of the form ay² + by + c = 0. Identifying coefficients a, b, and c correctly is essential to find the values of y in terms of x.

Algebraic Manipulation and Rearrangement

This concept involves rearranging terms, factoring, and simplifying expressions to isolate the desired variable. Careful manipulation is necessary to rewrite the equation in a standard quadratic form in y before applying the quadratic formula.

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