Frage: Löse nach \( x \): \( \frac{2}{x} + \frac{3}{x+1} = 1 \).
["Title: How to Solve the Equation ( \frac{2}{x} + \frac{3}{x+1} = 1 ) – Step-by-Step Guide", "Solving rational equations like ( \frac{2}{x} + \frac{3}{x+1} = 1 ) can seem challenging at first, but with the right approach, it’s completely manageable. If you're studying algebra or working through a math problem, especially asking „Frage: Löse nach ( x ): ( \frac{2}{x} + \frac{3}{x+1} = 1 )“, this guide will walk you through solving it clearly.", "---", "### Step-by-Step Solution", "We want to solve for ( x ) in:\n[\n\frac{2}{x} + \frac{3}{x+1} = 1\n]", "#### 1. Find a Common Denominator\nThe denominators are ( x ) and ( x+1 ). The least common denominator (LCD) is ( x(x+1) ). Use this to rewrite both fractions:", "[\n\frac{2(x+1)}{x(x+1)} + \frac{3x}{x(x+1)} = 1\n]", "#### 2. Combine the Fractions\nCombine the left-hand side over the common denominator:", "[\n\frac{2(x+1) + 3x}{x(x+1)} = 1\n]", "Simplify the numerator:\n[\n2x + 2 + 3x = 5x + 2\n]\nSo the equation becomes:\n[\n\frac{5x + 2}{x(x+1)} = 1\n]", "#### 3. Eliminate the Denominator\nMultiply both sides by ( x(x+1) ) (noting that ( x <br/>\neq 0 ) and ( x <br/>\neq -1 ) to avoid division by zero):", "[\n5x + 2 = x(x + 1)\n]", "#### 4. Expand and Rearrange into a Quadratic Equation\nExpand the right-hand side:\n[\n5x + 2 = x^2 + x\n]", "Move all terms to one side:\n[\n0 = x^2 + x - 5x - 2\n]\n[\nx^2 - 4x - 2 = 0\n]", "This is a quadratic equation in standard form:\n[\nx^2 - 4x - 2 = 0\n]", "#### 5. Solve the Quadratic Equation\nUse the quadratic formula ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ), where ( a = 1 ), ( b = -4 ), and ( c = -2 ):", "[\nx = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-2)}}{2(1)}\n= \frac{4 \pm \sqrt{16 + 8}}{2}\n= \frac{4 \pm \sqrt{24}}{2}\n]", "Simplify ( \sqrt{24} = \sqrt{4 \cdot 6} = 2\sqrt{6} ):\n[\nx = \frac{4 \pm 2\sqrt{6}}{2} = 2 \pm \sqrt{6}\n]", "So the solutions are:\n[\nx = 2 + \sqrt{6} \quad \ ext{and} \quad x = 2 - \sqrt{6}\n]", "---", "### 6. Check for Extraneous Solutions\nSince we multiplied by ( x(x+1) ), we must ensure neither solution makes a denominator zero.\n- ( x = 0 ): Not a solution.\n- ( x = -1 ): Not a solution.\nBoth ( 2 + \sqrt{6} ) and ( 2 - \sqrt{6} ) are valid because they don’t violate the domain.", "---", "### Final Note\nSolving rational equations requires careful manipulation and justification—especially checking extraneous roots. With careful steps like combining fractions, clearing denominators, and factoring, this equation yields elegant solutions. Practicing such problems improves algebraic fluency and confidence with rational expressions.", "---", "Keywords for SEO: \nSolveFrage x 2/x + 3/(x+1) = 1, how to solve rational equation, step-by-step rational equation solution, step-by-step solve ( \frac{2}{x} + \frac{3}{x+1} = 1 ), algebraic equation solving guide, quadratic formula application, rational expressions explained", "Meta Description:\nStep-by-step solution to ( \frac{2}{x} + \frac{3}{x+1} = 1 ). Learn to solve rational equations safely and accurately with clear examples and domain checks. Perfect for students mastering algebra."]
