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

Ch. 1 - Equations and Inequalities

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Lial 13th Edition

Ch. 1 - Equations and Inequalities

문제 81a

2장, 문제 81a

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

검증된 단계별 안내

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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주요 개념

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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.