\[ x^2 - 4x + 3 = 0 \] - United Radiology

["Solving the Quadratic Equation ( x^2 - 4x + 3 = 0 ): Step-by-Step Guide", "Understanding how to solve quadratic equations is a foundational skill in algebra and mathematics. One of the most commonly encountered problems is the equation:", "[
\nx^2 - 4x + 3 = 0
\n]", "Whether you're a student learning algebra or a teacher explaining key concepts, this article walks you through the best methods to solve ( x^2 - 4x + 3 = 0 ), explores its roots, and highlights its significance in mathematics.", "---", "### What Is a Quadratic Equation?", "A quadratic equation is any equation of the form:", "[
\nax^2 + bx + c = 0
\n]", "where ( a ), ( b ), and ( c ) are constants and ( a <br/>\neq 0 ). In our example:", "- ( a = 1 )
\n- ( b = -4 )
\n- ( c = 3 )", "Quadratic equations can be solved using several methods, including factoring, completing the square, and the quadratic formula. For this equation, factoring is simple and effective.", "---", "### Method 1: Factoring the Equation", "Factoring means rewriting the quadratic expression as a product of two binomials:", "[
\nx^2 - 4x + 3 = 0
\n]", "We look for two numbers that:", "- Multiply to ( c = 3 )
\n- Add up to ( b = -4 )", "The pair -1 and -3 satisfy these conditions:", "[
\n(-1) \ imes (-3) = 3 \quad \ ext{and} \quad (-1) + (-3) = -4
\n]", "So, the equation factors as:", "[
\n(x - 1)(x - 3) = 0
\n]", "To solve, set each factor equal to zero:", "[
\nx - 1 = 0 \quad \Rightarrow \quad x = 1
\n]
\n[
\nx - 3 = 0 \quad \Rightarrow \quad x = 3
\n]", "Solutions:
\n[
\n\boxed{x = 1 \quad \ ext{and} \quad x = 3}
\n]", "---", "### Method 2: Using the Quadratic Formula", "For any quadratic equation ( ax^2 + bx + c = 0 ), the solutions are given by:", "[
\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\n]", "Substitute ( 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 gives two solutions:", "- ( x = \frac{4 + 2}{2} = \frac{6}{2} = 3 )
\n- ( x = \frac{4 - 2}{2} = \frac{2}{2} = 1 )", "So again:
\n[
\n\boxed{x = 1 \quad \ ext{and} \quad x = 3}
\n]", "---", "### Method 3: Completing the Square", "Completing the square is another powerful technique. Start with:", "[
\nx^2 - 4x + 3 = 0
\n]", "Move the constant to the right:", "[
\nx^2 - 4x = -3
\n]", "To complete the square, take half of the coefficient of ( x ), square it, and add to both sides:", "(-4 \div 2 = -2), and ((-2)^2 = 4), so:", "[
\nx^2 - 4x + 4 = -3 + 4
\n]
\n[
\n(x - 2)^2 = 1
\n]", "Take square roots:", "[
\nx - 2 = \pm 1
\n]", "Solve for ( x ):", "[
\nx = 2 \pm 1
\n]", "Thus:
\n[
\nx = 3 \quad \ ext{or} \quad x = 1
\n]", "Consistent with earlier results.", "---", "### Why Solve ( x^2 - 4x + 3 = 0 )?", "While this equation may seem simple, mastering its solution forms the basis for tackling more complex quadratics. Applications include:", "- Physics: Modeling motion, projectile trajectories
\n- Economics: Profit and cost analysis
\n- Engineering: Designing parabolic structures
\n- Computer Science: Graphics and algorithm design", "Understanding the roots — here, ( x = 1 ) and ( x = 3 ) — reveals critical points in these models.", "---", "### Summary of Solutions", "For the equation ( x^2 - 4x + 3 = 0 ), the two real solutions are:", "[
\n\boxed{x = 1} \quad \ ext{and} \quad \boxed{x = 3}
\n]", "These roots can be interpreted graphically as the ( x)-intercepts of the parabola ( y = x^2 - 4x + 3 ), which opens upward because ( a = 1 > 0 ).", "---", "### Final Thoughts", "Mastering quadratic equations equips you with essential problem-solving skills used across science, engineering, and finance. Whether you use factoring, the quadratic formula, or completing the square, recognizing patterns quickly is key.", "Practice this equation and similar ones daily — soon, solving quadratics will feel intuitive, opening doors to advanced mathematics and real-world problem-solving.", "---", "Keywords: ( x^2 - 4x + 3 = 0 ), solve quadratic equation, factoring quadratic, quadratic formula, completing the square, algebra 2, math tutorial, quadratic roots, quadratic functions examples.", "Meta Description:
\nLearn how to solve ( x^2 - 4x + 3 = 0 ) using factoring, quadratic formula, and completing the square. Step-by-step guide with solutions and applications.", "---", "Related Reads:
\n- How to Graph a Quadratic Function
\n- Differences Between Linear and Quadratic Equations
\n- Common Quadratic Mistakes and How to Avoid Them", "---", "Stay curious. Master your algebra."]