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

Verified step by step guidance

1. Start with the given equation: \$4x^2 - 2xy + 3y^2 = 2$.
2. Rewrite the equation to isolate terms involving \(x\): \$4x^2 - 2xy = 2 - 3y^2$.
3. Recognize this as a quadratic equation in terms of \(x\): \$4x^2 - 2y x - (2 - 3y^2) = 0$.
4. Use the quadratic formula to solve for \(x\), where \(a = 4\), \(b = -2y\), and \(c = -(2 - 3y^2)\): \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
5. 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 \(y\).

Key Concepts

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.