2x^2 - 10x + 12 = 2(x^2 - 5x + 6) = 2(x - 2)(x - 3)

["# Solving the Quadratic Equation: 2x² – 10x + 12 = 2(x² – 5x + 6) and Its Factored Form 2(x – 2)(x – 3)", "When solving quadratic equations, rewriting expressions in factored form is often the key to simplifying work and uncovering roots quickly. One common technique involves transforming equations like 2x² – 10x + 12 = 0 into a product of binomial factors. In this article, we’ll explore how to rewrite 2x² – 10x + 12 as 2(x – 2)(x – 3), analyze its factorization, and understand how this form helps solve the equation effectively.", "## Why Factoring Helps", "Factoring a quadratic gives us a direct way to find the roots by setting each factor equal to zero. This method is faster and more intuitive than using the quadratic formula, especially for equations already structured in convenient forms. Factoring not only reveals solutions but also highlights key properties of the quadratic, such as its intercepts and symmetry.", "## Step-by-Step Transformation", "Start with the original quadratic expression:", "[\n2x^2 - 10x + 12\n]", "### Step 1: Factor out the greatest common factor (GCF)", "Both terms (2x², –10x, and 12) share a common factor of 2. Factor this out:", "[\n2(x^2 - 5x + 6)\n]", "### Step 2: Factor the quadratic inside the parentheses", "Now focus on factoring (x^2 - 5x + 6). We seek two numbers that multiply to 6 (the constant term) and add to –5 (the coefficient of x). These numbers are –2 and –3:", "[\nx^2 - 5x + 6 = (x - 2)(x - 3)\n]", "### Step 3: Substitute back into the expression", "Replace the quadratic, now fully factored, into the original expression:", "[\n2(x^2 - 5x + 6) = 2(x - 2)(x - 3)\n]", "Putting it all together:", "[\n2x^2 - 10x + 12 = 2(x - 2)(x - 3)\n]", "## Factored Form Interpretation", "From 2(x – 2)(x – 3), we see the equation becomes:", "[\n2(x - 2)(x - 3) = 0\n]", "This product equals zero only when one or more factors equals zero:", "- (x - 2 = 0 \implies x = 2)\n- (x - 3 = 0 \implies x = 3)", "Thus, the solutions to the equation are x = 2 and x = 3.", "## Benefits of This Factored Form", "- Easier solving: Finding roots is straightforward by solving each simple linear factor.\n- Understanding behavior: The roots x = 2 and x = 3 indicate where the parabola crosses the x-axis.\n- Graph insights: The vertex lies midway between the roots, providing symmetry and shape details.", "## Conclusion", "Rewriting 2x² – 10x + 12 as 2(x – 2)(x – 3) simplifies solving and deepens insight into the quadratic’s structure. By factoring out 2 and recognizing (x^2 - 5x + 6 = (x - 2)(x - 3)), we turn a standard quadratic into a product of binomials. This method is powerful for both solving and graphing, making it an essential tool for students and math enthusiasts alike.", "If you're tackling quadratics in algebra or calculus, learning to factor expressions like 2x² – 10x + 12 into 2(x – 2)(x – 3) ensures accuracy and speed—key advantages in both schoolwork and standardized tests.", "---", "Keywords:\nquadratic equation, factoring quadratics, factoring 2x² – 10x + 12, solving 2x² – 10x + 12, factored form 2(x – 2)(x – 3), roots of quadratic, algebra techniques, solving x² – 5x + 6, learning algebra factoring, quadratic solutions"]