["# Solving the Quadratic Equation: x² – 4x + 3 = 0", "Understanding how to solve quadratic equations is a fundamental skill in algebra, with wide applications in science, engineering, economics, and everyday problem-solving. One of the most commonly studied equations is x² – 4x + 3 = 0, a simple yet illustrative example of quadratic expressions. This article walks you through solving this equation step by step, explores its mathematical meaning, and highlights practical uses.", "---", "### What is a Quadratic Equation?", "A quadratic equation is any equation of the form:", "[ ax^2 + bx + c = 0 ]", "where:
\n- ( a ), ( b ), and ( c ) are constants, with ( a <br/>\neq 0 )
\n- ( x ) represents the unknown variable
\n- The term ( x^2 ) gives the equation a parabolic curve when graphed", "The equation ( x^2 - 4x + 3 = 0 ) fits this definition with:
\n- ( a = 1 )
\n- ( b = -4 )
\n- ( c = 3 )", "---", "### How to Solve ( x^2 - 4x + 3 = 0 )", "There are multiple methods to solve quadratic equations. The most common ones include factoring, completing the square, and using the quadratic formula. Let’s explore these in the context of this equation.", "#### 1. Factoring Method", "Factoring is often the quickest approach when the quadratic expresses easily into binomials.", "Step 1: Identify factors of the constant term (3) that sum to the coefficient of ( x ) (-4)", "We need two numbers whose product is ( 1 \ imes 3 = 3 ), and whose sum is ( -4 ). These numbers are ( -3 ) and ( -1 ):", "[
\nx^2 - 4x + 3 = (x - 3)(x - 1)
\n]", "Step 2: Apply the zero-product property", "Set each factor equal to zero:", "[
\nx - 3 = 0 \quad \ ext{or} \quad x - 1 = 0
\n]", "[
\nx = 3 \quad \ ext{or} \quad x = 1
\n]", "✅ The solutions are ( x = 1 ) and ( x = 3 ).", "---", "#### 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]", "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]", "So,", "[
\nx = \frac{4 + 2}{2} = 3 \quad \ ext{and} \quad x = \frac{4 - 2}{2} = 1
\n]", "Again, the solutions are ( x = 1 ) and ( x = 3 ).", "---", "#### 3. Graphical Interpretation", "The equation ( x^2 - 4x + 3 = 0 ) represents a parabola opening upwards (since the coefficient of ( x^2 ) is positive). The values of ( x ) where the graph intersects the x-axis are the real roots—precisely ( x = 1 ) and ( x = 3 ). These x-intercepts correspond to the solutions.", "---", "### What Do These Solutions Mean?", "- In algebraic terms, ( x = 1 ) and ( x = 3 ) are the values of ( x ) that make the expression equal to zero.
\n- In real-world models, such equations can represent break-even points, motion of objects along a path (via parabolic trajectories), or other optimization scenarios.", "---", "### Practical Applications of Quadratic Equations", "Studies of equations like ( x^2 - 4x + 3 = 0 ) extend beyond classroom exercises. Quadratic equations model:
\n- Projectile motion (where height over time is a quadratic function)
\n- Financial calculations involving profit maximization
\n- Geometry, such as finding dimensions of shapes with given area constraints
\n- Designing circuits and analyzing electrical systems", "Mastering these techniques strengthens analytical thinking and prepares learners for advanced math and science topics.", "---", "### Summary", "Solving ( x^2 - 4x + 3 = 0 ) reveals two clear solutions: ( x = 1 ) and ( x = 3 ). Through factoring or the quadratic formula, algebra equips us to decode such expressions efficiently. Whether you’re preparing for exams, solving real-world problems, or deepening your mathematical intuition, mastery of quadratic equations is essential.", "---", "Key takeaways:
\n- Factor ( x^2 - 4x + 3 ) into ( (x - 3)(x - 1) ) for easy solution finding
\n- Use the quadratic formula when factoring is difficult
\n- Visualizing the parabola reinforces the meaning of the roots
\n- Quadratic equations enable practical modeling across disciplines", "Start practicing today—solving ( x^2 - 4x + 3 = 0 ) is just the beginning!", "---", "References:
\n- Algebraic fundamentals on quadratic equations
\n- Applications of parabolas in physics and economics
\n- Step-by-step quadratic formula tutorials", "Keywords: x² – 4x + 3 = 0, quadratic equation solutions, factoring quadratic, quadratic formula, algebra practice, real-world applications, solving quadratics"]
