\[ (x - 1)(x^2 - 5x + 6) = 0 \] - United Radiology

["# Solving the Equation [(x - 1)(x^2 - 5x + 6) = 0]: Step-by-Step Guide", "If you’ve stumbled upon the equation
\n[ (x - 1)(x^2 - 5x + 6) = 0 ]
\nand are unsure how to solve it, you're in the right place. This article breaks down how to solve this equation using algebra fundamentals, explaining every step to help you understand solving quadratic and product equations. Plus, we’ll cover key concepts like solving polynomial equations, factoring quadratics, and the zero product property.", "---", "## What Does the Equation Mean?", "The equation
\n[ (x - 1)(x^2 - 5x + 6) = 0 ]
\nis a product of two factors equal to zero. Based on the Zero Product Property, if the product of factors equals zero, then at least one of the factors must be zero.", "So, to solve this equation, we set each factor equal to zero:
\n- ( x - 1 = 0 )
\n- ( x^2 - 5x + 6 = 0 )", "---", "## Step 1: Solve the Linear Factor", "Start with the simplest factor:
\n[ x - 1 = 0 ]", "Add 1 to both sides:
\n[ x = 1 ]", "✅ Solution 1: ( x = 1 )", "---", "## Step 2: Solve the Quadratic Factor", "Now solve:
\n[ x^2 - 5x + 6 = 0 ]", "This is a quadratic equation. Since it doesn’t factor trivially, we can factor it directly.", "Look for two numbers that multiply to ( +6 ) and add to ( -5 ).
\nThose numbers are ( -2 ) and ( -3 ):
\n[ x^2 - 5x + 6 = (x - 2)(x - 3) ]", "So, rewrite the equation:
\n[ (x - 1)(x - 2)(x - 3) = 0 ]", "Now apply the Zero Product Property again:
\n- ( x - 1 = 0 \Rightarrow x = 1 )
\n- ( x - 2 = 0 \Rightarrow x = 2 )
\n- ( x - 3 = 0 \Rightarrow x = 3 )", "✅ Solutions from the quadratic: ( x = 2 ) and ( x = 3 )", "---", "## Final Solutions", "Combining both results, the complete set of solutions to
\n[ (x - 1)(x^2 - 5x + 6) = 0 ]
\nis:
\n[
\n\boxed{x = 1,\ x = 2,\ \ ext{and } x = 3}
\n]", "---", "## Why Is This Equation Important?", "Understanding how to solve equations involving products helps in:
\n- Factoring polynomial equations
\n- Applying core concepts in algebra and calculus
\n- Modeling real-world problems where multiple conditions must be satisfied", "---", "## FAQ: Common Questions About ((x - 1)(x^2 - 5x + 6) = 0)", "Q: Why does setting the product to zero work?
\nA: This relies on the fundamental principle that if ( A \cdot B = 0 ), then ( A = 0 ) or ( B = 0 ) (or both). This allows solving equations by breaking them into simpler parts.", "Q: Can the quadratic ( x^2 - 5x + 6 = 0 ) be solved another way?
\nA: Yes! You could use the quadratic formula:
\n[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]
\nWith ( a = 1 ), ( b = -5 ), ( c = 6 ), the discriminant ( b^2 - 4ac = 25 - 24 = 1 ), so:
\n[ x = \frac{5 \pm 1}{2} \Rightarrow x = 3 \ ext{ and } x = 2 ]
\nwhich matches factoring.", "Q: Are there any repeated solutions?
\nA: In this equation, all roots are distinct: ( x = 1, 2, 3 ).", "---", "## Summary", "To solve [(x - 1)(x^2 - 5x + 6) = 0], apply the Zero Product Property by solving each factor:
\n- ( x - 1 = 0 \Rightarrow x = 1 )
\n- Factor the quadratic to get ( x = 2 ) and ( x = 3 )", "This approach is essential for solving polynomial equations by factoring and using key algebraic principles.", "---", "Keywords: solve ((x - 1)(x^2 - 5x + 6) = 0), quadratic equation solutions, zero product property, factoring polynomials, algebra tutorial, step-by-step equation solving, find x values, math help, polynomial roots, algebra fundamentals", "---", "Ready to practice? Try solving similar equations and reinforce your understanding of factoring and solving multi-factor equations!"]