Problem 5 - Algebra 1: Concepts and Skills

Question

Factor the expression. $$ w^{2}-16 w+64 $$

Step-by-Step Solution

Answer

The factored form of the given expression \(w^{2} - 16w + 64\) is \((w - 8)^{2}\)

Step 1: Identify the Structure and Terms

This expression is structured as a quadratic trinomial in the familiar form \(ax^{2}+bx+c\), where 'a', 'b', and 'c' are constants. In this problem, \(a = 1\), \(b = -16\), and \(c = 64\).

Step 2: Factoring the Quadratic Expression

We need to find two numbers that multiply to 64 (our 'c' value) and add up to -16 (our 'b' value). These two numbers will be -8 and -8, because \(-8 * -8 = 64\) and \(-8 + -8 = -16\). Therefore, the quadratic expression can be factored as \((w - 8)^{2}\)

Step 3: Checking the Result

To verify, expand \((w - 8)^{2}\) to see if it gives back the original expression. Applying the formula for the square of a binomial \((a - b)^{2} = a^{2} - 2ab + b^{2}\), where 'a' is w and 'b' is 8 in this case, we get the original expression \(w^{2} - 16w + 64\)