If \( x^2 - 5x + 6 = 0 \), what are the values of \( x \)?
["# Solving the Quadratic Equation ( x^2 - 5x + 6 = 0 ): Find All Values of ( x )", "When faced with a quadratic equation like ( x^2 - 5x + 6 = 0 ), solving for ( x ) becomes both essential and satisfying. Whether you're a student learning algebra or someone revisiting foundational math concepts, understanding how to determine the roots of a quadratic equation is valuable—especially since these values determine real-world applications such as projectile motion, economics, and optimization problems. In this article, we’ll walk through the step-by-step process of solving ( x^2 - 5x + 6 = 0 ) and reveal the precise values of ( x ) that satisfy the equation.", "## Understanding the Equation", "The general form of a quadratic equation is:", "[\nax^2 + bx + c = 0\n]", "Comparing this with our given equation ( x^2 - 5x + 6 = 0 ), we identify the coefficients:", "- ( a = 1 )\n- ( b = -5 )\n- ( c = 6 )", "Since ( a <br/>\neq 0 ), this is indeed a quadratic equation. The solutions for ( x ) can be found using factoring, completing the square, or the famous quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "For this equation, factoring is especially straightforward because the quadratic is simple and factors nicely.", "## Solving by Factoring", "We look for two numbers that multiply to ( c = 6 ) and add up to ( b = -5 ).", "The numbers ( -2 ) and ( -3 ) work because:", "[\n(-2) \ imes (-3) = 6 \quad \ ext{and} \quad (-2) + (-3) = -5\n]", "Thus, we factor the equation as:", "[\nx^2 - 5x + 6 = (x - 2)(x - 3) = 0\n]", "Setting each factor equal to zero gives:", "[\nx - 2 = 0 \quad \Rightarrow \quad x = 2\n]\n[\nx - 3 = 0 \quad \Rightarrow \quad x = 3\n]", "## Confirming with the Quadratic Formula", "For completeness, we confirm our solutions using the quadratic formula:", "[\nx = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(6)}}{2(1)} = \frac{5 \pm \sqrt{25 - 24}}{2} = \frac{5 \pm \sqrt{1}}{2}\n]", "This simplifies to:", "[\nx = \frac{5 \pm 1}{2}\n]", "So,", "[\nx = \frac{5 + 1}{2} = 3 \quad \ ext{and} \quad x = \frac{5 - 1}{2} = 2\n]", "Both methods yield the same solutions: ( x = 2 ) and ( x = 3 ).", "## Summary of Values", "The values of ( x ) that satisfy the equation ( x^2 - 5x + 6 = 0 ) are:", "[\n\boxed{x = 2 \quad \ ext{and} \quad x = 3}\n]", "## Why These Roots Matter", "The solutions ( x = 2 ) and ( x = 3 ) are not just numbers—they pinpoint key moments or points in many real-life scenarios. For instance:", "- In business, these may represent break-even points or maximum profit levels.\n- In physics, they could mark the time when a projectile reaches certain heights.\n- In geometry, they may represent intersection points of curves or lines.", "Understanding where quadratic equations cross the x-axis—i.e., where they evaluate to zero—provides insight into behavior and change.", "## Final Thoughts", "Solving ( x^2 - 5x + 6 = 0 ) is a classic example of applying algebraic techniques to unlock meaningful results. Whether you use factoring for speed or the quadratic formula for precision, the solutions consistently lead to ( x = 2 ) and ( x = 3 ). Mastering this process empowers you to tackle more advanced math with confidence.", "Remember: quadratic equations are not just abstract symbols—they model situations and reveal answers hidden in equations. The values of ( x ) that satisfy ( x^2 - 5x + 6 = 0 ) are:", "[\n\boxed{2 \quad \ ext{and} \quad 3}\n]", "Happy solving!"]
