+ 24x + 18x + 4x² = 180

["Solving the Quadratic Equation: +24x + 18x + 4x² = 180", "If you’ve stumbled across the equation 24x + 18x + 4x² = 180, you’re likely looking to solve for x—a common challenge in algebra. This expression combines linear terms with a quadratic term, forming a standard quadratic equation that can be efficiently solved using algebraic techniques. In this article, we’ll break down how to simplify, rewrite, and solve 4x² + 42x = 180, providing clear steps and practical insights for learners and math enthusiasts alike.", "---", "### Step 1: Simplify the Linear Terms", "Start by combining like terms on the left-hand side:", "[\n24x + 18x + 4x^2 = 4x^2 + (24x + 18x) = 4x^2 + 42x\n]", "So the equation becomes:", "[\n4x^2 + 42x = 180\n]", "---", "### Step 2: Rewrite in Standard Quadratic Form", "To solve for x, move all terms to one side to form a standard quadratic equation (at least equal to zero):", "[\n4x^2 + 42x - 180 = 0\n]", "---", "### Step 3: Simplify the Equation (Optional but Recommended)", "Before applying the quadratic formula, simplify the equation by dividing every term by the greatest common divisor (GCD) of the coefficients. Here, GCD of 4, 42, and 180 is 2:", "[\n\frac{4x^2}{2} + \frac{42x}{2} - \frac{180}{2} = 0 \quad \Rightarrow \quad 2x^2 + 21x - 90 = 0\n]", "Now, solving 2x² + 21x - 90 = 0 is simpler and more efficient.", "---", "### Step 4: Apply the Quadratic Formula", "Use the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Where:\n- ( a = 2 )\n- ( b = 21 )\n- ( c = -90 )", "Calculate the discriminant:", "[\n\Delta = b^2 - 4ac = 21^2 - 4(2)(-90) = 441 + 720 = 1161\n]", "Now compute:", "[\nx = \frac{-21 \pm \sqrt{1161}}{2 \cdot 2} = \frac{-21 \pm \sqrt{1161}}{4}\n]", "Since 1161 is not a perfect square, the solutions are irrational:", "[\nx = \frac{-21 + \sqrt{1161}}{4} \quad \ ext{and} \quad x = \frac{-21 - \sqrt{1161}}{4}\n]", "---", "### Step 5: Approximate the Solutions (Optional)", "To get numerical values:", "[\n\sqrt{1161} \approx 34.07\n]", "So:", "[\nx_1 \approx \frac{-21 + 34.07}{4} = \frac{13.07}{4} \approx 3.27\n]\n[\nx_2 \approx \frac{-21 - 34.07}{4} = \frac{-55.07}{4} \approx -13.77\n]", "---", "### Summary & Key Takeaways", "- The equation 4x² + 42x = 180 simplifies to 2x² + 21x - 90 = 0 by dividing by 2.\n- Use the quadratic formula to solve:\n [\n x = \frac{-21 \pm \sqrt{1161}}{4}\n ]\n- Solutions are irrational but can be approximated numerically.\n- Simplifying equations before solving saves time and reduces error risk.", "---", "### Why This Equation Matters", "Quadratic equations like 4x² + 42x = 180 appear in real-world applications—from projectile motion to optimization problems. Mastering their solution builds critical analytical skills for advanced math, science, and engineering fields.", "---", "Keywords: quadratic equation, solve 4x² + 42x = 180, simplify quadratic, quadratic formula, algebra solution, 24x + 18x + 4x² = 180, step-by-step math, rational solutions, irrational numbers in algebra.", "---", "Want to solve more equations? Keep practicing—each quadratic brings you one step closer to mathematical fluency!"]