Divide the equation by 3: \( x^2 - 4x + 3 = 0 \).

["# Dividing the Quadratic Equation ( x^2 - 4x + 3 = 0 ) by 3: A Step-by-Step Guide", "When solving quadratic equations, simplifying them can make finding roots and understanding their behavior easier. One common step is dividing the entire equation by a constant—especially in the case of ( x^2 - 4x + 3 = 0 ), where dividing by 3 offers a streamlined form.", "## Understanding the Original Equation", "The equation ( x^2 - 4x + 3 = 0 ) is a standard quadratic in the form ( ax^2 + bx + c = 0 ), with coefficients:", "- ( a = 1 )\n- ( b = -4 )\n- ( c = 3 )", "This equation can be solved using factoring, completing the square, or the quadratic formula. However, dividing the entire equation by 3 offers a quick way to simplify it before applying any method.", "## How to Divide the Equation by 3", "To divide the equation ( x^2 - 4x + 3 = 0 ) by 3, simply apply division to every term:", "[\n\frac{x^2}{3} - \frac{4x}{3} + \frac{3}{3} = 0\n]", "This simplifies to:", "[\n\frac{1}{3}x^2 - \frac{4}{3}x + 1 = 0\n]", "### Why Divide by 3?", "Dividing by 3 transforms the equation into a simplified version where all coefficients are fractions. While this doesn’t change the roots of the equation, it often makes the quadratic formula easier to apply or the completing-the-square method clearer:", "[\n\frac{1}{3}x^2 - \frac{4}{3}x + 1 = 0\n]", "Multiply every term by 3 again to eliminate the fractions (optional):", "[\nx^2 - 4x + 3 = 0\n]", "---", "## Solving the Simplified Equation", "### Using the Quadratic Formula", "The quadratic formula is ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).\nUsing our original equation ( x^2 - 4x + 3 = 0 ), plug in ( a = 1 ), ( b = -4 ), ( c = 3 ):", "[\nx = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(3)}}{2(1)} = \frac{4 \pm \sqrt{16 - 12}}{2} = \frac{4 \pm \sqrt{4}}{2} = \frac{4 \pm 2}{2}\n]", "This yields two solutions:", "[\nx = \frac{4 + 2}{2} = 3 \quad \ ext{and} \quad x = \frac{4 - 2}{2} = 1\n]", "---", "## Benefits of Dividing the Equation by 3", "- Simplifies coefficients: Working with fractions like ( \frac{1}{3} ) is easier in symbolic manipulation.\n- Enhances readability and computation: A cleaner form often reduces errors, especially in hand calculations.\n- Maintains equivalence: The roots remain unchanged—in this case, ( x = 1 ) and ( x = 3 )—so the equation stays exact.", "---", "## When to Divide by a Constant", "Before dividing by a coefficient in a quadratic equation:", "- Confirm it does not introduce division by zero or undefined behavior.\n- Ensure dividing maintains the equation’s algebraic equivalence.\n- Use dividing by 3 when simplifying does not complicate further solving methods.", "---", "## Conclusion", "Dividing ( x^2 - 4x + 3 = 0 ) by 3 transforms the equation into a simplified, equivalent form:\n[\n\frac{1}{3}x^2 - \frac{4}{3}x + 1 = 0\n]", "This step efficiently prepares the quadratic for standard solution techniques and streamlines calculations. Whether using factoring, the quadratic formula, or graphical methods, dividing by 3 enhances clarity and precision in solving ( x^2 - 4x + 3 = 0 ).", "---", "Keywords: divide quadratic equation by 3, solve ( x^2 - 4x + 3 = 0 ), simplifying quadratic, quadratic formula, equation solution tips, algebra homework, quadratic simplification.\nMeta Description: Learn how to divide ( x^2 - 4x + 3 = 0 ) by 3, simplify the equation, and solve the quadratic with ease using algebra steps and the quadratic formula."]