Solution:** For the quadratic equation \( x^2 - 5x + 6 = 0 \), Vieta's formulas tell us that the product of the roots is given by \( \frac{d}{a} \), where \( a \) is the coefficient of \( x^2 \) and \( d \) is the constant term. Here, \( a = 1 \) and \( d = 6 \), so the product of the roots is:

["Solving the Quadratic Equation ( x^2 - 5x + 6 = 0 ): Understanding Vieta’s Formulas", "Learning how to solve quadratic equations is a fundamental skill in algebra, and one powerful way to understand their roots without full factoring is through Vieta’s formulas. These mathematical truths provide direct relationships between the coefficients of a quadratic equation and its roots, making problem-solving more efficient and insightful.", "---", "### The Quadratic Equation in Standard Form", "A standard quadratic equation is expressed as:\n[\nax^2 + bx + c = 0\n]\nwhere ( a ), ( b ), and ( c ) are coefficients, and ( a <br/>\ne 0 ).", "For the equation:\n[\nx^2 - 5x + 6 = 0\n]\nwe identify the coefficients:\n- ( a = 1 ) (the coefficient of ( x^2 ))\n- ( b = -5 ) (the coefficient of ( x ))\n- ( c = 6 ) (the constant term)", "---", "### Vieta’s Formulas: Connection Between Coefficients and Roots", "Vieta’s formulas reveal key relationships between the roots and the coefficients of a quadratic equation. If ( r_1 ) and ( r_2 ) are the two roots, then:", "[\nr_1 + r_2 = -\frac{b}{a} \quad \ ext{(sum of roots)}\n]\n[\nr_1 \cdot r_2 = \frac{c}{a} \quad \ ext{(product of roots)}\n]", "The product of the roots is particularly useful because it depends only on the ratio of the constant term ( c ) to the leading coefficient ( a ).", "---", "### Applying Vieta’s Formula to the Given Equation", "Given ( a = 1 ) and ( c = 6 ), Vieta’s formula for the product of the roots becomes:\n[\nr_1 \cdot r_2 = \frac{c}{a} = \frac{6}{1} = 6\n]", "This means that regardless of how you solve for ( r_1 ) and ( r_2 )—whether by factoring, completing the square, or using the quadratic formula—the product of the solutions will always equal 6.", "---", "### Why This Matters", "Understanding this product helps verify solutions:\nIf you factor ( x^2 - 5x + 6 = 0 ) into ( (x - 2)(x - 3) = 0 ), the roots are ( x = 2 ) and ( x = 3 ).\nCalculating the product: ( 2 \cdot 3 = 6 ), which matches Vieta’s result.", "Moreover, Vieta’s formulas allow quick checking without fully computing roots. This is especially valuable in advanced math, physics, and engineering contexts where efficiency and accuracy are crucial.", "---", "### Summary", "For the quadratic equation ( x^2 - 5x + 6 = 0 ):\n[\n\ ext{Product of roots} = \frac{d}{a} = \frac{6}{1} = 6\n]\nSo, the roots multiply to 6 — a foundational insight that deepens understanding and strengthens problem-solving skills.", "Whether you're a student mastering algebra or a professional applying mathematical models, leveraging Vieta’s relationships simplifies learning and enhances precision.", "---", "Keywords for SEO:\nquadratic equation Vieta’s formulas, solve ( x^2 - 5x + 6 = 0 ), product of roots Vieta, algebraic verification quadratic, quadratic relationships, Vieta sum and product roots, algebra tutorial quadratic, find product of roots Vieta x²-5x+6", "---", "Understanding Vieta’s formulas not only accelerates solving equations but also reveals deeper mathematical structures—making every quadratic a gateway to broader algebraic insight."]