Question: Solve for $ a $: $ rac{a^2 - 4a + 3}{a - 1} = 2 $.

["# Solve for $ a $: $ \dfrac{a^2 - 4a + 3}{a - 1} = 2 $", "Understanding how to solve rational equations is essential for mastering algebra. One common problem students encounter is solving equations like:", "$$\n\dfrac{a^2 - 4a + 3}{a - 1} = 2\n$$", "This equation involves a quadratic expression divided by a linear term, set equal to a constant. In this article, we’ll walk through the step-by-step solution, simplify the expression, and discuss important algebraic concepts such as domain restrictions and factoring.", "---", "## Step 1: Simplify the Numerator", "The first step is to factor the numerator:\n$$\na^2 - 4a + 3\n$$", "We look for two numbers that multiply to $ 3 $ and add to $ -4 $. These numbers are $ -1 $ and $ -3 $. So, the numerator factors as:\n$$\na^2 - 4a + 3 = (a - 1)(a - 3)\n$$", "Now rewrite the equation:\n$$\n\dfrac{(a - 1)(a - 3)}{a - 1} = 2\n$$", "---", "## Step 2: Cancel Common Factors", "The expression $ \dfrac{a - 1}{a - 1} $ simplifies to $ 1 $, but only when $ a <br/>\ne 1 $ because division by zero is undefined.", "So, for $ a <br/>\ne 1 $, we can simplify:\n$$\na - 3 = 2\n$$", "---", "## Step 3: Solve the Simplified Equation", "Solve:\n$$\na - 3 = 2 \quad \Rightarrow \quad a = 5\n$$", "---", "## Step 4: Check for Extraneous Solutions", "Recall from Step 1 that $ a = 1 $ makes the original denominator zero, which is not allowed. Our solution is $ a = 5 $, which is valid since it doesn’t violate the domain restriction.", "---", "## Final Answer", "$$\n\boxed{a = 5}\n$$", "---", "## Why This Problem Matters", "Solving equations of the form $ \dfrac{P(a)}{Q(a)} = k $ teaches students:", "- How to factor quadratic expressions\n- How to simplify rational expressions\n- How to identify restrictions on variables to avoid undefined expressions\n- The importance of checking solutions", "This problem is a great example of applying algebraic skills to rational equations — fundamental for higher-level math like calculus and advanced algebra.", "---", "### Summary", "- Factor numerator: $ a^2 - 4a + 3 = (a - 1)(a - 3) $\n- Simplify: $ \dfrac{(a - 1)(a - 3)}{a - 1} = a - 3 $, for $ a <br/>\ne 1 $\n- Solve: $ a - 3 = 2 $ → $ a = 5 $\n- Confirm $ a = 5 $ is valid (not excluded by denominator)", "Understanding how to solve such equations breaks down complex rational forms into manageable steps—boosting confidence and accuracy in algebra!", "---", "Keywords: solve $ \dfrac{a^2 - 4a + 3}{a - 1} = 2 $, algebra step-by-step, solving rational equations, simplify rational expressions, domain restrictions in algebra, how to solve for $ a $ in rational equations, step-by-step quadratic equation, equation solving tips, algebra homework help.", "---", "Read more: Learn how to solve quadratic rational equations, explore factoring techniques, practice advanced algebra problems, and improve your equation-solving skills today!"]