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Chapter 1: Problem 79

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

Short Answer

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

Step by step solution

01

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.
02

Determine the Terms

The operations between the terms are all additions or subtractions. The terms of the quadratic are \(x^2\), \(-6x\), and \(9\).
03

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 \((x-3)(x-3)\).
04

Simplify

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

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Most popular questions from this chapter

Solve equation by the method of your choice. $$ \frac{x-1}{x-2}+\frac{x}{x-3}=\frac{1}{x^{2}-5 x+6} $$

In Exercises 95鈥�102, use interval notation to represent all values of x satisfying the given conditions. $$ y_{1}=\frac{2}{3}(6 x-9)+4, y_{2}=5 x+1, \text { and } y_{1}>y_{2} $$

If a quadratic equation has imaginary solutions, how is this shown on the graph of \(y=a x^{2}+b x+c ?\)

Explain how to solve \(x^{2}+6 x+8=0\) by completing the square.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I prefer interval notation over set-builder notation because it takes less space to write solution sets.
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