Solve each equation. See Example 2. 3x²/(x-1) + 2 = x/(x-1)

Verified step by step guidance

1. Identify the given equation: \(\frac{3x^2}{x-1} + 2 = \frac{x}{x-1}\).
2. To eliminate the denominators, multiply every term in the equation by the common denominator, which is \((x-1)\), to clear the fractions.
3. After multiplying, simplify each term: \$3x^2 + 2(x-1) = x$.
4. Distribute the 2 on the left side: \$3x^2 + 2x - 2 = x$.
5. Bring all terms to one side to set the equation equal to zero: \$3x^2 + 2x - 2 - x = 0\(, which simplifies to \)3x^2 + x - 2 = 0$.

Key Concepts

Rational Equations

Rational equations involve expressions with variables in the denominator. Solving them requires finding a common denominator to combine terms or eliminate fractions, then solving the resulting polynomial equation. It's important to check for restrictions where the denominator equals zero.

Domain Restrictions

Domain restrictions arise from values that make denominators zero, which are undefined in rational expressions. Identifying these values prevents invalid solutions. Before solving, determine and exclude these values to ensure the solution set is valid.

Solving Quadratic Equations

After clearing denominators, the equation often reduces to a quadratic form. Solving quadratic equations can be done by factoring, completing the square, or using the quadratic formula. Understanding these methods is essential to find all possible solutions.