Problem 45 - Elementary Algebra

Question

For exercises 1-66, simplify. $$ \frac{x^{2}-4 x+3}{x^{2}-3 x+2} $$

Step-by-Step Solution

Answer

\( \frac{x-3}{x-2} \)

Step 1: Factor the numerator

The numerator of the expression is \(x^2 - 4x + 3\). Factor this quadratic expression. We look for two numbers that multiply to 3 (the constant term) and add to -4 (the coefficient of the linear term). These numbers are -1 and -3. Therefore, \(x^2 - 4x + 3\) factors to \((x-1)(x-3)\).

Step 2: Factor the denominator

The denominator of the expression is \(x^2 - 3x + 2\). Factor this quadratic expression by finding two numbers that multiply to 2 (the constant term) and add to -3 (the coefficient of the linear term). These numbers are -1 and -2. Thus, \(x^2 - 3x + 2\) factors to \((x-1)(x-2)\).

Step 3: Rewrite the expression with factored terms

Substitute the factored forms of the numerator and denominator back into the original fraction. This gives: \[ \frac{(x-1)(x-3)}{(x-1)(x-2)} \]

Step 4: Simplify the expression

Cancel out the common factor \((x-1)\) in both the numerator and the denominator. The simplified expression is: \( \frac{x-3}{x-2} \)

Key Concepts

Factoring Quadratics

Factoring quadratics means breaking down a quadratic expression into simpler expressions that multiply together to give the original quadratic. In our exercise, we started with the quadratic expressions: \(x^2 - 4x + 3\) and \(x^2 - 3x + 2\). To factor these, we need to find two numbers that meet two conditions:

• They multiply to the constant term (without the variable x).
• They add up to the coefficient of the middle term (the term with x).

For the numerator \(x^2 - 4x + 3\), we need two numbers that multiply to 3 and add up to -4. These numbers are -1 and -3. So, we write the numerator as \((x-1)(x-3)\). For the denominator \(x^2 - 3x + 2\), we want two numbers that multiply to 2 and add up to -3. These numbers are -1 and -2, so the denominator factors to \((x-1)(x-2)\). Factoring makes complex algebraic expressions more manageable and easier to simplify.

Common Factors

Common factors are terms that appear in both the numerator and the denominator. Identifying common factors is essential for simplification. After factoring both the numerator and the denominator in our example exercise, we get \(\frac{(x-1)(x-3)}{(x-1)(x-2)}\). Here, \((x-1)\) is a common factor. Removing common factors helps us simplify the expression. Think of it like reducing fractions in arithmetic; we remove the same factors from the top and bottom. When we cancel out the common factor \((x-1)\) from both the numerator and the denominator, we are left with \(\frac{x-3}{x-2}\). Always remember, canceling out common factors is a fundamental step for simplifying algebraic expressions.

Simplification Process

The simplification process involves reducing an expression to its simplest form. This usually requires factoring and canceling common factors. Here is a step-by-step simplification process we followed in our example:

• First, factor both the numerator and the denominator. We got \((x-1)(x-3)\) and \((x-1)(x-2)\) respectively.
• Next, rewrite the original expression using these factored forms, resulting in \(\frac{(x-1)(x-3)}{(x-1)(x-2)}\).
• Then, identify and cancel any common factors. In this case, \((x-1)\) is the common factor in both the numerator and the denominator.
• Finally, rewrite the simplified expression. After canceling \((x-1)\), we were left with \(\frac{x-3}{x-2}\).

Following each step carefully ensures you don't miss any details and results in the correct, simplified form. This process not only applies to our example but also to other algebraic expressions needing simplification.