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

Verified step by step guidance

1. Start by identifying the given equation: $4x^2 - 2xy + 3y^2 = 2$.
2. Rearrange the equation to isolate terms involving $x$: $4x^2 - 2xy = 2 - 3y^2$.
3. Factor out $x$ from the left side: $x(4x - 2y) = 2 - 3y^2$.
4. Solve for $x$ by dividing both sides by $(4x - 2y)$: $x = \frac{2 - 3y^2}{4x - 2y}$.
5. Simplify the expression if possible, keeping $x$ in terms of $y$.

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0. In the context of the given equation, it can be rearranged to fit this form, allowing us to apply methods such as factoring, completing the square, or using the quadratic formula to solve for x in terms of y.

Solving for a Variable

Solving for a variable involves isolating that variable on one side of the equation. In this case, we need to manipulate the equation to express x solely in terms of y. This may require rearranging terms, applying algebraic operations, and potentially using the quadratic formula to find the values of x that correspond to a given y.

Discriminant

The discriminant of a quadratic equation, given by the formula D = b^2 - 4ac, determines the nature of the roots of the equation. It indicates whether the solutions are real and distinct, real and repeated, or complex. Understanding the discriminant is crucial when solving for x in terms of y, as it informs us about the possible values of x based on the value of y.