\[ 3(x - 3)(x - 1) = 0 \] - United Radiology

["Understanding the Equation 3(x - 3)(x - 1) = 0: A Complete Guide", "Mathematics often presents problems in the form of equations that might initially seem complex but reveal elegant solutions once broken down. One such straightforward yet fundamental equation is:", "[
\n3(x - 3)(x - 1) = 0
\n]", "Whether you're a student learning algebra, a teacher explaining key principles, or simply someone curious about equations, this article breaks down the solution process, reveals the underlying math, and explores real-world relevance.", "---", "### What Makes This Equation Important?", "At first glance, the equation looks simple—a product of constants and binomial factors equals zero. However, solving it demonstrates core algebraic techniques such as the Zero Product Property, factoring quadratics, and understanding solution sets. More than just finding values, solving this equation teaches how to approach equations systematically.", "---", "### Step-by-Step Solution", "The equation:
\n[
\n3(x - 3)(x - 1) = 0
\n]", "follows the principle of the Zero Product Property, which states that if the product of several factors equals zero, then at least one of the factors must be zero.", "Step 1: Apply the Zero Product Property
\nSet each factor equal to zero:", "[
\n3 = 0 \quad \ ext{(Not possible — discard this)}
\n]
\n[
\nx - 3 = 0 \quad \Rightarrow \quad x = 3
\n]
\n[
\nx - 1 = 0 \quad \Rightarrow \quad x = 1
\n]", "Since 3 ≠ 0, we discard that factor. The valid solutions are:", "[
\nx = 3 \quad \ ext{and} \quad x = 1
\n]", "---", "### Solutions and Their Meaning", "The equation ( 3(x - 3)(x - 1) = 0 ) has two distinct real solutions:
\n- ( x = 1 )
\n- ( x = 3 )", "These values represent the roots or zeros of the associated quadratic expression. Geometrically, they are the x-intercepts of the parabola defined by ( y = 3(x - 3)(x - 1) ). Since this is a quadratic equation in factored form, the graph is a parabola that opens upwards (due to the positive leading coefficient after expansion).", "---", "### Expanding (Optional but Insightful)", "To deepen understanding, expanding the equation:", "[
\n3(x - 3)(x - 1) = 3\left[(x \cdot x) - (x \cdot 1) - (3 \cdot x) + (3 \cdot 1)\right]
\n= 3(x^2 - x - 3x + 3)
\n= 3(x^2 - 4x + 3)
\n= 3x^2 - 12x + 9
\n]", "So,
\n[
\n3x^2 - 12x + 9 = 0
\n]", "Using the quadratic formula:
\n[
\nx = \frac{12 \pm \sqrt{(-12)^2 - 4 \cdot 3 \cdot 9}}{2 \cdot 3} = \frac{12 \pm \sqrt{144 - 108}}{6} = \frac{12 \pm \sqrt{36}}{6} = \frac{12 \pm 6}{6}
\n]", "Thus:
\n[
\nx = \frac{18}{6} = 3 \quad \ ext{and} \quad x = \frac{6}{6} = 1
\n]", "(Consistent with earlier results.)", "---", "### Why Are These Solution Methods Valuable?", "- Factorization: Recognizing and applying the Zero Product Property simplifies solutions without calculus or advanced tools.
\n- Vertex and Axis of Symmetry: The roots ( x = 1 ) and ( x = 3 ) mean the axis of symmetry is at ( x = 2 ), key for graphing.
\n- Applications: These roots can represent critical points in physics (motion), economics (break-even analysis), or engineering (optimization).", "---", "### Teaching and Studying Tips", "- Visualize: Graph ( y = 3(x - 3)(x - 1) ) to see intercepts at ( x = 1 ) and ( x = 3 ).
\n- Practice: Rewrite expanded forms ( 3x^2 - 12x + 9 = 0 ) to check equivalence.
\n- Deepen Concepts: Explore how changing coefficients affects roots — sensitivity of solutions to parameters.", "---", "### Conclusion", "The equation ( 3(x - 3)(x - 1) = 0 ) might look simple, but it encapsulates essential algebraic reasoning and problem-solving skills. By applying the Zero Product Property, factoring, and expanding, learners not only solve for ( x ) but also build a strong foundation for higher-level math. Whether used in class, exam prep, or self-study, mastering this equation unlocks broader mathematical insight.", "---", "Keywords for SEO:
\n3(x – 3)(x – 1) = 0, solve quadratic equations, algebra basics, factoring equation, zero product property, quadratic roots, math explanation, solving equations step-by-step, how to find x in polynomial equations, real roots, graphing parabolas", "---", "Understanding equations like ( 3(x - 3)(x - 1) = 0 ) transforms abstract symbols into meaningful solutions — the heart of mathematical thinking."]