- 48x - 30x + 4x^2 = 252

["Solving the Equation: 48x – 30x + 4x² = 252 – A Step-by-Step Guide", "Mathematics often involves solving equations to uncover unknown values — a fundamental skill in both academic and real-world applications. One such equation that regularly appears in algebra courses and problem-solving sessions is:", "[\n48x - 30x + 4x^2 = 252\n]", "This article will guide you through simplifying and solving this quadratic equation step-by-step, helping you understand how to manage linear and quadratic terms efficiently. Whether you're a student learning algebra or a educator explaining quadratic solutions, this breakdown will clarify the process and boost confidence in handling similar equations.", "---", "### Step 1: Simplify the Left Side of the Equation", "Start by combining like terms on the left-hand side (LHS). Notice that ( 48x - 30x ) are like terms since they both contain variable ( x ):", "[\n48x - 30x = 18x\n]", "Now rewrite the equation:", "[\n18x + 4x^2 = 252\n]", "Next, to write it in standard quadratic form, rearrange terms with the highest degree first:", "[\n4x^2 + 18x - 252 = 0\n]", "This is a standard quadratic equation in the form:", "[\nax^2 + bx + c = 0\n]", "where:\n( a = 4 ),\n( b = 18 ),\n( c = -252 ).", "---", "### Step 2: Simplify the Equation Further", "To make solving easier, divide every term by the greatest common divisor (GCD) of the coefficients. Here, GCD(4, 18, 252) = 2:", "[\n\frac{4x^2 + 18x - 252}{2} = \frac{0}{2}\n]", "This simplifies cleanly to:", "[\n2x^2 + 9x - 126 = 0\n]", "Now the equation is in simplified quadratic form, ready for solution.", "---", "### Step 3: Solve the Quadratic Equation", "To solve ( 2x^2 + 9x - 126 = 0 ), we use the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Plug in ( a = 2 ), ( b = 9 ), ( c = -126 ):", "- Calculate discriminant (( b^2 - 4ac )):\n[\n9^2 - 4(2)(-126) = 81 + 1008 = 1089\n]", "- Take the square root of discriminant:\n[\n\sqrt{1089} = 33\n]", "- Substitute into the formula:", "[\nx = \frac{-9 \pm 33}{2 \cdot 2} = \frac{-9 \pm 33}{4}\n]", "This yields two solutions:", "1. ( x = \frac{-9 + 33}{4} = \frac{24}{4} = 6 )\n2. ( x = \frac{-9 - 33}{4} = \frac{-42}{4} = -10.5 )", "---", "### Step 4: Check The Solutions", "Plug ( x = 6 ) and ( x = -10.5 ) back into the original equation to verify:", "For ( x = 6 ):\n[\n48(6) - 30(6) + 4(6)^2 = 288 - 180 + 144 = 252 \quad \ ext{✓}\n]", "For ( x = -10.5 ):\n[\n48(-10.5) - 30(-10.5) + 4(-10.5)^2 = -504 + 315 + 441 = 252 \quad \ ext{✓}\n]", "Both solutions are valid.", "---", "### Why This Problem Is Important", "Solving equations like ( 48x - 30x + 4x^2 = 252 ) builds essential algebraic skills, including:", "- Simplifying algebraic expressions\n- Combining like terms\n- Standardizing quadratic forms\n- Applying the quadratic formula\n- Verifying solutions", "Understanding these steps also prepares students for more advanced math topics like graphing, optimization, and real-world modeling.", "---", "### Final Answer", "The solutions to the equation ( 48x - 30x + 4x^2 = 252 ) are:", "[\n\boxed{ x = 6 \quad \ ext{and} \quad x = -10.5 }\n]", "---", "Tips for mastering similar problems:\n- Always simplify equations before solving.\n- Use the quadratic formula when factoring is difficult.\n- Double-check solutions by substituting back into the original equation.\n- Practice a variety of quadratic equations to build speed and confidence.", "If you enjoyed this step-by-step algebraic guide, explore more equations and techniques at Online Algebra Tutoring — master math with confidence!", "---", "Keywords: solve 48x – 30x + 4x² = 252, quadratic equation solutions, algebraic simplification, quadratic formula guide, 4x² + 18x – 252 = 0, step-by-step math problem solving."]