Problem 79 - Algebra and Trigonometry

Question

Exercises \(78-80\) will help you prepare for the material covered in the next section. Factor: \(x^{2}-6 x+9\)

Step-by-Step Solution

Answer

The factored form of the equation \(x^{2} - 6x + 9\) is \((x-3)^2\).

Step 1: Recognize the Structure

The given equation is a quadratic of the form \(ax^2 + bx + c\). This type of equation can usually be factored. The goal is to rewrite the equation in a different way to simplify it.

Step 2: Determine the Terms

The operations between the terms are all additions or subtractions. The terms of the quadratic are \(x^2\), \(-6x\), and \(9\).

Step 3: Factor the Quadratic

In order to factor, we need to find two numbers that both add up to \(-6\) (the coefficient of the \(x\) term) and multiply to \(9\). The numbers that satisfy this are \(-3\) and \(-3\), hence, we can factor the equation to \((x-3)(x-3)\).

Step 4: Simplify

Because both factors are identical, the factored equation can be simplified as \((x-3)^2\)