For each equation, solve for x in terms of y. 2x2 + 4xy - 3y2 = 2

Ch. 1 - Equations and Inequalities

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

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Lial 13th Edition

Ch. 1 - Equations and Inequalities

Problem 81a

Chapter 2, Problem 81a

For each equation, solve for x in terms of y. 2x² + 4xy - 3y² = 2

Verified step by step guidance

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

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

Recognize this as a quadratic equation in $x$: $2x^2 + 4yx - (2 + 3y^2) = 0$.

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

Substitute the values of $a$, $b$, and $c$ into the formula and simplify under the square root to express $x$ in terms of $y$.

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Key Concepts

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

Solving Quadratic Equations

A quadratic equation is a polynomial equation of degree two, typically in the form ax² + bx + c = 0. To solve for x, you can use methods such as factoring, completing the square, or the quadratic formula. In this problem, x appears in a quadratic form, so applying the quadratic formula is essential to express x in terms of y.

Treating Variables as Parameters

When solving for one variable in terms of another, treat the other variable as a constant or parameter. Here, y is considered a constant while solving for x. This approach allows you to rewrite the equation as a quadratic in x with coefficients depending on y, enabling the use of standard solution techniques.

Quadratic Formula

The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) provides the solutions to any quadratic equation ax² + bx + c = 0. Identifying a, b, and c correctly when the coefficients depend on y is crucial. This formula helps find explicit expressions for x in terms of y, including both possible roots.

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