Problem 48 - Elementary Algebra

Question

Factor completely. $$ (x-4) 2+5(x-4)+6 $$

Step-by-Step Solution

Answer

The expression factors to \((x - 2)(x - 1)\).

Step 1: Identify Common Terms

Let's start by recognizing the expression and calling the common factor. Notice that the expression can be rewritten for clarity. Let us denote \( u = x-4 \). The expression becomes \( u^2 + 5u + 6 \).

Step 2: Factoring the Quadratic Expression

Now, we need to factor \( u^2 + 5u + 6 \). We are looking for two numbers that multiply to 6 (the constant term) and add up to 5 (the coefficient of the linear term). The numbers 2 and 3 satisfy these requirements because \( 2 \times 3 = 6 \) and \( 2 + 3 = 5 \).

Step 3: Write the Factored Expression

Using the numbers found in Step 2, we can factor the quadratic as \( (u + 2)(u + 3) \). Substitute back \( u = x - 4 \) into this factored form, yielding \((x - 4 + 2)(x - 4 + 3)\).

Step 4: Simplify the Expression

Finally, simplify the expression to get \((x - 2)(x - 1)\). The complete factorization of the original expression \((x-4)^2 + 5(x-4) + 6\) is \((x - 2)(x - 1)\).

Key Concepts

Quadratic Expression

A quadratic expression is a polynomial of degree two, and it generally takes the form: \[ ax^2 + bx + c \] Here, "a", "b", and "c" are constants where "a" is not equal to zero. In the problem we're looking at, the quadratic expression initially appears as \((x-4)^2 + 5(x-4) + 6\). To simplify tackling quadratic expressions, we often use substitutions. In our exercise, we used \( u = x-4 \), which transforms the expression into \( u^2 + 5u + 6 \). This method helps us focus directly on the quadratic form, making it easier to factor or simplify.

Common Factor

The common factor is a term that is shared by two or more terms in an expression. Identifying a common factor is useful because it allows us to rewrite or simplify the expression. In the initial exercise, the expression \( (x-4)^2 + 5(x-4) + 6 \) shared a common component of \( x-4 \). By substituting, we simplified the given expression into \( u^2 + 5u + 6 \) where \( u \) represented the common factor \( x-4 \).

• This method enables us to break down complex expressions into simpler parts.
• Noting a common factor can make factoring a quadratic expression much more straightforward.

Simplification

Simplification is the process of reducing an expression to a simpler or more manageable form. It's often the final step after factoring, as it presents the expression in a clean and concise format. Once we factored the expression using substitution, as shown before, we arrived at \((u + 2)(u + 3)\). After substituting back \( u = x-4 \), it became \((x-4 + 2)(x-4 + 3)\). Simplifying this gives us the final result: \((x-2)(x-1)\).

• Always return to the original variables after using substitution.
• Ensure no further reduction is possible to confirm the expression is fully simplified.

This simplification ensures we have the neatest version of the factorized quadratic expression, ready to be used or solved further.