Solve each equation. 2 - 5/x = 3/x²

Verified step by step guidance

1. Start with the given equation: \(2 - \frac{5}{x} = \frac{3}{x^{2}}\).
2. Identify the least common denominator (LCD) to eliminate the fractions. The denominators are \(x\) and \(x^{2}\), so the LCD is \(x^{2}\).
3. Multiply every term in the equation by \(x^{2}\) to clear the denominators: \(x^{2} \cdot 2 - x^{2} \cdot \frac{5}{x} = x^{2} \cdot \frac{3}{x^{2}}\).
4. Simplify each term after multiplication: \$2x^{2} - 5x = 3$.
5. Rewrite the equation in standard quadratic form by moving all terms to one side: \$2x^{2} - 5x - 3 = 0$.

Key Concepts

Rational Equations

Rational equations involve expressions with variables in the denominator. Solving them requires finding a common denominator or eliminating denominators by multiplying both sides by the least common denominator (LCD) to simplify the equation.

Least Common Denominator (LCD)

The LCD is the smallest expression that all denominators in a rational equation can divide into evenly. Identifying the LCD allows you to clear fractions by multiplying both sides of the equation, making it easier to solve.

Checking for Extraneous Solutions

When solving rational equations, some solutions may make denominators zero, which are invalid. After solving, substitute solutions back into the original equation to ensure they do not cause division by zero.