Multipliez par \( v(v + 20) \) : \( 300(v + 20) + 300v = 7v(v + 20) \).
["Title: Simplifying the Equation: Solving ( 300(v + 20) + 300v = 7v(v + 20) )", "When faced with algebraic equations, understanding how to simplify and solve expressions step-by-step is essential. In this SEO-optimized guide, we break down the equation:", "[\n300(v + 20) + 300v = 7v(v + 20)\n]", "### Understanding the Equation", "The left side combines two terms with a common factor of 300:", "[\n300(v + 20) + 300v = 300v + 6000 + 300v = 600v + 6000\n]", "The right side multiplies ( 7v ) by ( (v + 20) ):", "[\n7v(v + 20) = 7v^2 + 140v\n]", "Now our equation becomes:", "[\n600v + 6000 = 7v^2 + 140v\n]", "### Rearranging to Standard Quadratic Form", "Move all terms to one side to form a standard quadratic equation:", "[\n0 = 7v^2 + 140v - 600v - 6000\n]", "Simplify:", "[\n0 = 7v^2 - 460v - 6000\n]", "### Solving the Quadratic Equation", "Use the quadratic formula:", "[\nv = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Here, ( a = 7 ), ( b = -460 ), and ( c = -6000 ). Plugging in:", "[\nv = \frac{460 \pm \sqrt{(-460)^2 - 4(7)(-6000)}}{2(7)}\n]", "Calculate discriminant:", "[\n(-460)^2 = 211600,\quad 4 \cdot 7 \cdot 6000 = 168000\n]", "[\n\sqrt{211600 + 168000} = \sqrt{379600} = 616\n]", "Thus:", "[\nv = \frac{460 \pm 616}{14}\n]", "This gives two solutions:", "[\nv = \frac{460 + 616}{14} = \frac{1076}{14} = 77,\quad \ ext{so } v = 77\n]", "[\nv = \frac{460 - 616}{14} = \frac{-156}{14} = -\frac{78}{7} \approx -11.14\n]", "### Summary", "The equation ( 300(v + 20) + 300v = 7v(v + 20) ) simplifies and transforms into the quadratic equation ( 7v^2 - 460v - 6000 = 0 ). Solving it yields two real solutions:", "- ( v = 77 )\n- ( v = -\frac{78}{7} )", "### Why This Equation Matters", "Understanding how to expand, simplify, and solve expressions like ( 300(v + 20) + 300v = 7v(v + 20) ) helps in modeling real-world scenarios involving scaling, cost functions, or proportional relationships. Mastering algebraic manipulation also strengthens critical thinking for advanced STEM studies.", "Keywords: algebraic equation, simplify expression, solve quadratic, 300(v + 20) + 300v = 7v(v + 20), step-by-step solving, linear and quadratic equations, educational math tutorial, mathematical problems, algebra practice, factoring equations, solving for ( v )", "Meta Description: Simplify and solve the equation ( 300(v + 20) + 300v = 7v(v + 20) ) step-by-step using standard algebraic techniques. Learn how to manipulate expressions, rearrange quadratics, and find real solutions. Ideal for students and learners tackling intermediate algebra."]
