4x^2 + 70x + 250 = 600
["Solving the Equation: 4x² + 70x + 250 = 600 — Step-by-Step Guide", "If you’ve come across the quadratic equation 4x² + 70x + 250 = 600, you’re not alone. Solving such equations is a fundamental skill in algebra and essential for mastering math at school or in real-world applications like engineering, finance, and data analysis. This article breaks down how to rewrite, simplify, and solve this equation step by step, making it easy to understand for students, educators, and math enthusiasts alike.", "---", "### What Is the Equation?", "Start with the quadratic equation:", "[\n4x^2 + 70x + 250 = 600\n]", "First, to solve for (x), we bring all terms to one side to form a standard quadratic equation set to zero.", "---", "### Step 1: Simplify the Equation", "Subtract 600 from both sides:", "[\n4x^2 + 70x + 250 - 600 = 0\n]", "[\n4x^2 + 70x - 350 = 0\n]", "Now we have the standard form:", "[\n4x^2 + 70x - 350 = 0\n]", "---", "### Step 2: Simplify Further (Optional but Smart)", "Since all coefficients are even, divide the entire equation by 2 to make calculations easier:", "[\n2x^2 + 35x - 175 = 0\n]", "This reduction simplifies the next steps without changing the solutions.", "---", "### Step 3: Use the Quadratic Formula", "The general form of a quadratic equation is:", "[\nax^2 + bx + c = 0\n]", "For our simplified equation:", "- (a = 2),\n- (b = 35),\n- (c = -175)", "The quadratic formula is:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "---", "### Step 4: Calculate the Discriminant", "First compute the discriminant ((D)):", "[\nD = b^2 - 4ac = 35^2 - 4(2)(-175) = 1225 + 1400 = 2625\n]", "Since (D > 0), there are two distinct real solutions.", "---", "### Step 5: Compute the Square Root", "[\n\sqrt{D} = \sqrt{2625}\n]", "Factor 2625 to simplify:", "[\n2625 = 25 \ imes 105 = 25 \ imes 3 \ imes 35 = 25 \ imes 3 \ imes 5 \ imes 7\n]", "[\n\sqrt{2625} = \sqrt{25 \ imes 105} = 5\sqrt{105}\n]", "This exact form is useful, but for decimal approximation:", "[\n\sqrt{2625} \approx 51.24\n]", "---", "### Step 6: Plug Into the Quadratic Formula", "[\nx = \frac{-35 \pm \sqrt{2625}}{2 \ imes 2} = \frac{-35 \pm 51.24}{4}\n]", "---", "### Step 7: Calculate Both Solutions", "1. First solution:", "[\nx = \frac{-35 + 51.24}{4} = \frac{16.24}{4} = 4.06\n]", "2. Second solution:", "[\nx = \frac{-35 - 51.24}{4} = \frac{-86.24}{4} = -21.56\n]", "---", "### Final Answer", "The solutions to the equation 4x² + 70x + 250 = 600 are approximately:", "[\n\boxed{x \approx 4.06} \quad \ ext{and} \quad \boxed{x \approx -21.56}\n]", "---", "### Why Solve This Equation?", "Beyond the numbers, understanding how to solve such equations helps you:", "- Interpret real-world scenarios (like projectile motion or profit calculations)\n- Model data in science and economics\n- Prepare for advanced math topics like calculus or systems of equations\n- Improve problem-solving and logical thinking", "---", "### Summary", "- Rewrite the equation in standard form\n- Simplify if possible (by dividing by common factors)\n- Apply the quadratic formula carefully\n- Use exact or approximate values depending on context", "By mastering these steps, you gain confidence not only in solving quadratics but also in approaching complex mathematical challenges.", "---", "Keywords:\nquadratic equation, solve 4x² + 70x + 250 = 600, step-by-step solution, quadratic formula, algebra basics, solving equations, math tutorial, real-world math applications", "---", "If you found this guide helpful, share it with classmates or bookmark it for future problem-solving needs!"]
