Find the roots of the quadratic equation \( x^2 - 5x + 6 = 0 \) using factoring.
["# Find the Roots of the Quadratic Equation ( x^2 - 5x + 6 = 0 ) Using Factoring", "Solving quadratic equations is a fundamental skill in algebra, and one of the most straightforward methods is factoring. In this article, we’ll walk through the process of finding the roots of the quadratic equation ( x^2 - 5x + 6 = 0 ) by factoring. Understanding how to break down and solve quadratic equations opens the door to mastering more complex algebraic concepts.", "## What is a Quadratic Equation?", "A quadratic equation is any equation of the form:", "[\nax^2 + bx + c = 0\n]", "where ( a ), ( b ), and ( c ) are constants and ( a <br/>\ne 0 ). In our case,", "[\nx^2 - 5x + 6 = 0\n]", "has coefficients ( a = 1 ), ( b = -5 ), and ( c = 6 ).", "## Why Factoring?", "Factoring involves expressing the quadratic expression as a product of two binomials. Once factored, setting each binomial equal to zero allows us to solve for ( x ) easily. Factoring works best when the quadratic expression is easily decomposable into integer products.", "## Step-by-Step: Factoring ( x^2 - 5x + 6 )", "### Step 1: Identify coefficients\nWe want two numbers that:", "- Multiply to ( c = 6 )\n- Add up to ( b = -5 )", "### Step 2: Find the factor pairs of 6\nThe pairs of integers whose product is 6 are:\n- ( 1 \ imes 6 )\n- ( 2 \ imes 3 )\n- ( (-1) \ imes (-6) )\n- ( (-2) \ imes (-3) )", "### Step 3: Choose the pair that adds to -5\nAmong these, the pair ( -2 ) and ( -3 ) satisfies:", "[\n-2 + (-3) = -5 \quad \ ext{and} \quad (-2) \cdot (-3) = 6\n]", "### Step 4: Rewrite the middle term using the factor pair\nNow rewrite the original trinomial using ( -2 ) and ( -3 ):", "[\nx^2 - 2x - 3x + 6 = 0\n]", "### Step 5: Group and factor by grouping\nGroup the terms:", "[\n(x^2 - 2x) + (-3x + 6) = 0\n]", "Factor out the common terms:", "[\nx(x - 2) - 3(x - 2) = 0\n]", "### Step 6: Factor out the common binomial\nBoth terms contain ( (x - 2) ), so factor it out:", "[\n(x - 2)(x - 3) = 0\n]", "## Step 7: Apply the Zero Product Property", "Set each factor equal to zero:", "[\nx - 2 = 0 \quad \Rightarrow \quad x = 2\n]", "[\nx - 3 = 0 \quad \Rightarrow \quad x = 3\n]", "## Final Answer: The Roots", "The solutions — or roots — of the quadratic equation ( x^2 - 5x + 6 = 0 ) are:", "[\n\boxed{x = 2 \quad \ ext{and} \quad x = 3}\n]", "## Why This Method Works", "Factoring transforms the equation into a product of simpler expressions, making it easy to find values of ( x ) that satisfy the equation. For equations where factoring is not intuitive, alternative methods like the quadratic formula or completing the square can be used, but factoring remains a powerful and efficient tool when applicable.", "## Summary", "- Factoring breaks the quadratic into simpler binomials.\n- Identify two numbers multiplying to ( c ) and adding to ( b ).\n- Rewrite and group terms to factor by grouping.\n- Set each binomial equal to zero to find the roots.", "Mastering factoring gives you a strong foundation for solving quadratic equations and extends your algebraic abilities. Practice with different quadratics to become fluent—the roots will become second nature!", "---", "Keywords: quadratic equation ( x^2 - 5x + 6 = 0 ), factoring, solving quadratics, roots of a quadratic, algebra 1, factoring trinomials, algebra tutorial, solving quadratic using factoring, quadratic roots."]
