A quadratic equation x² - 5x + 6 = 0 has roots. Find the roots.
["## Solving the Quadratic Equation: x² - 5x + 6 = 0 and Finding Its Roots", "Understanding quadratic equations is a fundamental part of algebra, and solving them opens doors to countless applications in science, engineering, and mathematics. One commonly encountered equation is ( x^2 - 5x + 6 = 0 ). In this article, we’ll explore how to find the roots of this quadratic equation step-by-step, making the process clear and easy to follow.", "### What is a Quadratic Equation?", "A quadratic equation is a polynomial equation of degree 2, generally written in the form:", "[\nax^2 + bx + c = 0\n]", "where ( a ), ( b ), and ( c ) are constants, and ( a <br/>\neq 0 ). The roots of the equation are the values of ( x ) that satisfy the equation — essentially, the solutions where the expression equals zero.", "---", "### Given Equation:\n[\nx^2 - 5x + 6 = 0\n]\nHere, ( a = 1 ), ( b = -5 ), and ( c = 6 ).", "---", "### Step 1: Choose a Method to Solve", "There are three primary methods to find the roots of a quadratic equation:", "- Factoring\n- Using the quadratic formula\n- Completing the square", "For this equation, factoring is the most straightforward approach.", "---", "### Step 2: Factor the Quadratic Expression", "We want to express ( x^2 - 5x + 6 ) as a product of two binomials:", "[\nx^2 - 5x + 6 = (x - m)(x - n)\n]", "We look for two numbers ( m ) and ( n ) such that:\n- ( m \ imes n = +6 ) (the constant term)\n- ( m + n = -5 ) (the coefficient of ( x ))", "The numbers (-2) and (-3) satisfy these conditions because:\n- ((-2) \ imes (-3) = 6)\n- ((-2) + (-3) = -5)", "Thus, the equation factors as:", "[\n(x - 2)(x - 3) = 0\n]", "---", "### Step 3: Solve for the Roots", "Set each factor equal to zero:", "[\nx - 2 = 0 \quad \Rightarrow \quad x = 2\n]\n[\nx - 3 = 0 \quad \Rightarrow \quad x = 3\n]", "---", "### Final Results", "The roots of the quadratic equation ( x^2 - 5x + 6 = 0 ) are:\n[\n\boxed{x = 2} \quad \ ext{and} \quad \boxed{x = 3}\n]", "---", "### Verification", "To double-check, substitute ( x = 2 ) and ( x = 3 ) back into the original equation:", "1. For ( x = 2 ):\n[\n2^2 - 5(2) + 6 = 4 - 10 + 6 = 0\n]\n2. For ( x = 3 ):\n[\n3^2 - 5(3) + 6 = 9 - 15 + 6 = 0\n]", "Both values satisfy the equation, confirming they are correct roots.", "---", "### Conclusion", "Solving quadratic equations like ( x^2 - 5x + 6 = 0 ) is efficient when using factoring. By identifying the right pair of numbers, we factor the quadratic and find the roots simply by setting each factor to zero. This method not only provides an elegant solution but also reinforces core algebraic skills essential for advanced mathematical problem-solving.", "Whether you're studying algebra for school, preparing for standardized tests, or tackling real-world modeling problems, mastering how to find roots of quadratic equations will strengthen your analytical abilities—making you better equipped to handle complex equations with confidence."]
