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

Verified step by step guidance

1. Start by writing down the given equation: \(\frac{3x^{2}}{x-1} + 2 = \frac{x}{x-1}\).
2. Identify the common denominator on both sides, which is \(x-1\). To eliminate the denominators, multiply every term in the equation by \(x-1\).
3. After multiplying, simplify each term: \(\left(\frac{3x^{2}}{x-1}\right)(x-1) + 2(x-1) = \left(\frac{x}{x-1}\right)(x-1)\), which simplifies to \$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$. This is a quadratic equation ready to be solved.

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 values that make denominators zero, as these are excluded from the solution.

Domain Restrictions

Domain restrictions arise from values that make any denominator zero, which are undefined in the equation. Identifying these values before solving prevents including invalid solutions. For example, if the denominator is (x - 1), then x cannot be 1.

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. Solutions must be checked against domain restrictions to ensure validity.