Combinez les termes : \( 600v + 6000 = 7v^2 + 140v \).

["Title: Solve the Quadratic Equation: 600v + 6000 = 7v² + 140v – Step-by-Step Guide", "—\nDiscover how to solve the equation ( 600v + 6000 = 7v^2 + 140v ) efficiently with clear steps. Whether you're a student tackling algebra or just seeking a better understanding, this guide explains how to simplify and solve this quadratic equation. We’ll also highlight digital tools and exact formation of terms like combining like terms and rearranging into standard quadratic form.", "---", "### Understanding the Equation: Combine Like Terms & Rearrange Terms", "The equation begins with:\n[\n600v + 6000 = 7v^2 + 140v\n]", "Step 1: Move all terms to one side to form a standard quadratic equation\nReorganize by subtracting ( 600v + 6000 ) from both sides:", "[\n0 = 7v^2 + 140v - 600v - 6000\n]", "Now simplify the right-hand side by combining like terms:", "- ( 140v - 600v = -460v )", "So the equation becomes:", "[\n7v^2 - 460v - 6000 = 0\n]", "---", "### Combine Quadratic Terms Clearly", "The equation ( 7v^2 - 460v - 6000 = 0 ) is now in standard quadratic form:\n[\nav^2 + bv + c = 0\n]\nwhere\n- ( a = 7 )\n- ( b = -460 )\n- ( c = -6000 )", "Such organization enhances readability and eases computation—key when solving complex expressions like combining terms and applying the quadratic formula.", "---", "### Solve Using the Quadratic Formula", "To solve ( 7v^2 - 460v - 6000 = 0 ), use the quadratic formula:", "[\nv = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Step 2: Plug in values\n[\nv = \frac{-(-460) \pm \sqrt{(-460)^2 - 4(7)(-6000)}}{2 \cdot 7}\n]", "[\nv = \frac{460 \pm \sqrt{211600 + 168000}}{14}\n]", "[\nv = \frac{460 \pm \sqrt{379600}}{14}\n]", "---", "### Simplify the Square Root Term", "Simplify ( \sqrt{379600} ):", "Check perfect squares nearby:\n( 616^2 = 379456 ), and ( 617^2 = 38089 ), so ( \sqrt{379600} ) is not a perfect square. We keep it as ( \sqrt{379600} ), or simplify:", "[\n379600 = 100 \ imes 3796 = 4 \ imes 94900\n\Rightarrow \sqrt{379600} = 2\sqrt{94900}\n]", "(Use calculator in practice for exact decimal value.)", "---", "### Final Computation", "Approximate ( \sqrt{379600} \approx 614.94 )", "So,", "[\nv = \frac{460 \pm 614.94}{14}\n]", "Two solutions:", "1. ( v = \frac{460 + 614.94}{14} = \frac{1074.94}{14} \approx 76.85 )\n2. ( v = \frac{460 - 614.94}{14} = \frac{-154.94}{14} \approx -11.07 )", "---", "### Summary: Why Combining Terms Matters", "Combining like terms (e.g., ( 140v - 600v = -460v )) simplifies the equation and reduces errors during rearrangement. This step is essential before applying the quadratic formula and ensures accuracy in your solution.", "---", "### Bonus: Tools & Resources to Confirm Solutions", "- Use graphing calculators or apps like Desmos or Wolfram Alpha to verify solutions\n- Check plugging values back into original equation ( 600v + 6000 = 7v^2 + 140v )\n- Practice similar equations to strengthen algebraic fluency", "---", "Conclusion:\nMastering steps like combining terms, simplifying quadratics, and using the quadratic formula turns complex equations into manageable steps. Whether solving for v or understanding algebraic structures, clear organization enhances your math skills and confidence. Solve confidently — the equation ( 600v + 6000 = 7v^2 + 140v ) leads to two real roots, proving quadratic equations yield insight when approached systematically.", "---", "Keywords: solve quadratic equation, 600v + 6000 = 7v² + 140v, combine like terms, quadratic formula, solve 7v² – 460v – 6000 = 0, algebraic equation solving, step-by-step algebra, quadratic solutions, math tips for students."]