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

Ch. 1 - Equations and Inequalities

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Lial 13th Edition

Ch. 1 - Equations and Inequalities

문제 79a

2장, 문제 79a

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

검증된 단계별 안내

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

Rewrite the equation to isolate terms involving \(x\): \(4x^2 - 2xy = 2 - 3y^2\).

Recognize this as a quadratic equation in terms of \(x\): \(4x^2 - 2y x - (2 - 3y^2) = 0\).

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

Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula and simplify under the square root to express \(x\) explicitly 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 ax^2 + bx + c = 0. To solve for x, you can use methods such as factoring, completing the square, or the quadratic formula. In this problem, since the equation involves both x and y, treat y as a constant and solve the quadratic in terms of x.

Treating Variables as Parameters

When solving for one variable in terms of another, consider the other variable as a constant or parameter. Here, y is treated as a known value, allowing the equation to be viewed as a quadratic in x. This approach helps isolate x and express it explicitly in terms of y.

Using the Quadratic Formula

The quadratic formula x = [-b ± sqrt(b^2 - 4ac)] / (2a) provides solutions to any quadratic equation ax^2 + bx + c = 0. Identify coefficients a, b, and c with respect to x, substitute them into the formula, and simplify to express x in terms of y. This method is essential when factoring is difficult or impossible.