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. To eliminate the denominators, multiply every term in the equation by \(x^{2}\), which is the least common denominator (LCD) of the fractions.
3. After multiplying, rewrite each term: \$2 \cdot x^{2} - 5 \cdot x = 3$.
4. Rearrange the equation to standard quadratic form: \$2x^{2} - 5x - 3 = 0$.
5. Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\) with \(a=2\), \(b=-5\), and \(c=-3\) to find the values of \(x\).

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.

Finding a Common Denominator

To combine or clear fractions in an equation, identify the least common denominator (LCD) of all rational expressions. Multiplying both sides of the equation by the LCD eliminates denominators, simplifying the equation to a polynomial form that is easier to solve.

Solving Quadratic Equations

After clearing denominators, the equation often becomes quadratic. Solving quadratic equations involves methods like factoring, completing the square, or using the quadratic formula to find variable values. Always verify solutions in the original equation to exclude extraneous roots.