Expand and Solve:** \( 250 + 50x + 20x + 4x^2 = 400 \) → \( 4x^2 + 70x + 250 - 400 = 0 \) → \( 4x^2 + 70x - 150 = 0 \)

["Expand and Solve: Mastering the Quadratic Equation ( 4x^2 + 70x - 150 = 0 )", "Are you struggling to solve quadratic equations with multiple terms like ( 250 + 50x + 20x + 4x^2 )? Simplifying expressions and solving quadratic equations can seem discouraging at first—but with the right steps, this process becomes straightforward and empowering.", "In this article, we’ll walk through how to expand, simplify, and solve the quadratic equation ( 4x^2 + 70x - 150 = 0 ), transforming it from a complex-looking expression into a solvable form.", "---", "### Step 1: Combine Like Terms", "The original expression\n[ 250 + 50x + 20x + 4x^2 ]\ncan be simplified by combining like terms:\n- Constants: ( 250 )\n- Linear terms: ( 50x + 20x = 70x )\n- Quadratic term: ( 4x^2 )", "This gives:\n[ 4x^2 + 70x + 250 ]", "---", "### Step 2: Set the Equation to Zero", "To solve for ( x ), we move all terms to one side:\n[\n4x^2 + 70x + 250 = 400\n]\nSubtract 400 to form a standard quadratic equation:\n[\n4x^2 + 70x + 250 - 400 = 0 \implies 4x^2 + 70x - 150 = 0\n]\nThis matches the simplified form we’re solving:\n[\n4x^2 + 70x - 150 = 0\n]", "---", "### Step 3: Simplify the Equation (Optional but Helpful)", "Before applying any root-finding method, check if the equation can be simplified by dividing all terms by their greatest common divisor. Here, every coefficient (4, 70, -150) is divisible by 2:", "[\n\frac{4x^2 + 70x - 150}{2} = 0 \implies 2x^2 + 35x - 75 = 0\n]", "While not strictly necessary, simplifying reduces error risk in later steps—especially when solving using the quadratic formula.", "---", "### Step 4: Use the Quadratic Formula to Solve", "Quadratic equations of the form ( ax^2 + bx + c = 0 ) are best solved using the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "For our equation ( 4x^2 + 70x - 150 = 0 ):\n- ( a = 4 )\n- ( b = 70 )\n- ( c = -150 )", "Plug values into the formula:", "[\nx = \frac{-70 \pm \sqrt{70^2 - 4(4)(-150)}}{2(4)}\n]", "Calculate the discriminant (( D = b^2 - 4ac )):\n[\n70^2 = 4900\n\quad \ ext{and} \quad\n4 \cdot 4 \cdot (-150) = -2400\n\Rightarrow D = 4900 - (-2400) = 4900 + 2400 = 7300\n]", "So,\n[\nx = \frac{-70 \pm \sqrt{7300}}{8}\n]", "Simplify ( \sqrt{7300} ):\n( 7300 = 100 \ imes 73 ), so\n[\n\sqrt{7300} = \sqrt{100 \ imes 73} = 10\sqrt{73}\n]", "Thus:\n[\nx = \frac{-70 \pm 10\sqrt{73}}{8}\n]", "Reduce the fraction by dividing numerator and denominator by 2:\n[\nx = \frac{-35 \pm 5\sqrt{73}}{4}\n]", "---", "### Step 5: Interpret the Solutions", "The equation ( 4x^2 + 70x - 150 = 0 ) has two real solutions:\n[\nx = \frac{-35 + 5\sqrt{73}}{4} \quad \ ext{and} \quad x = \frac{-35 - 5\sqrt{73}}{4}\n]", "These irrational roots reflect the quadratic’s nature—two distinct real values of ( x ) satisfying the original equation.", "---", "### Why This Matters", "Learning to expand complex expressions and solve quadratics builds foundational math skills essential in physics, engineering, economics, and computer science. Using tools like the quadratic formula—while understanding the algebra behind it—opens doors to deeper problem-solving confidence.", "---", "### Pro Tips for Solving Quadratics", "- Always combine like terms first.\n- Move all terms to one side to achieve ( = 0 ) form.\n- Simplify coefficients when possible.\n- Identify ( a ), ( b ), and ( c ) clearly.\n- Always compute the discriminant—its sign determines the root type.\n- Verify solutions by plugging back into the original equation.", "---", "### Final Thoughts", "What began as a messy expression:\n[ 250 + 50x + 20x + 4x^2 ]\nevolved into the solvable quadratic equation:\n[ 4x^2 + 70x - 150 = 0 ]\nnow stands solved and fully understood.", "With practice and clarity in each step, quadratic equations lose their complexity—and become powerful tools for real-world problem solving.", "---", "Keywords:\nquadratic equation solution, expand and solve, solve 4x² + 70x - 150 = 0, quadratic formula tutorial, simplify algebraic expressions, solve real quadratic equations, expand والأس solve 4x² + 70x - 150 = 0, step-by-step quadratic solve, algebra help with quadratics", "Meta Description:\nLearn how to expand ( 250 + 50x + 20x + 4x^2 ), simplify to ( 4x^2 + 70x + 250 ), and solve the quadratic equation ( 4x^2 + 70x - 150 = 0 ) using step-by-step simplification and the quadratic formula. Perfect for students advancing algebra skills.", "---", "Don’t stop here—master substitution, factoring, and the full spectrum of quadratic solutions for complete confidence in algebra!"]