How to Subtract Rational Expressions: GED Math Tutorial

Subtracting Rational Expressions

• 🎯 The goal is to subtract two rational expressions: 4x + 2 and 2x / (x - 3).
• 🔑 To subtract rational expressions, you must first find a common denominator.
• 🛠️ Multiply the first fraction by (x + 2) / (x + 2) and the second fraction by (x - 3) / (x - 3) to achieve common denominators.

Simplifying the Expression

• 📈 After distribution, the numerator of the first expression becomes (4x + 2)(x + 2), resulting in 4x^2 + 8x + 2x + 4, which simplifies to 4x^2 + 10x + 4.
• 📉 The numerator of the second expression becomes 2x(x - 3), resulting in 2x^2 - 6x.
• ⚠️ Crucially, distribute the negative sign from the subtraction to both terms of the second numerator: - (2x^2 - 6x) becomes -2x^2 + 6x.
• ➕ Combine the numerators: (4x^2 + 10x + 4) - (2x^2 - 6x) becomes 4x^2 + 10x + 4 - 2x^2 + 6x.
• 🧮 Simplify the combined numerator: (4x^2 - 2x^2) + (10x + 6x) + 4 results in 2x^2 + 16x + 4.
• ➡️ The final simplified expression is (2x^2 + 16x + 4) / ((x - 3)(x + 2)).

Common Denominator Method (Alternative Approach)

• 🔄 An alternative method involves multiplying the first fraction by (x+2)/(x+2) and the second by (x-3)/(x-3).
• ➕ The first numerator becomes (4x+2)(x+2) = 4x^2 + 8x + 2x + 4 = 4x^2 + 10x + 4.
• ➖ The second numerator becomes 2x(x-3) = 2x^2 - 6x.
• ⚠️ Remember to distribute the negative sign: -(2x^2 - 6x) = -2x^2 + 6x.
• ➕ Combine numerators: 4x^2 + 10x + 4 - 2x^2 + 6x.
• 🧮 Simplify: 2x^2 + 16x + 4.
• ➡️ The final result is (2x^2 + 16x + 4) / ((x - 3)(x + 2)).

Final Answer

• ✅ The calculation leads to the expression -2x^2 - 12 over the common denominator (x - 3)(x + 2).
• 💡 The terms 4x and -4x cancel out during simplification, leaving -2x^2 - 12 as the final numerator.