Problem 12 - Beginning and Intermediate Algebra

Question

Factor completely. $$q^{2}-22 q+121$$

Step-by-Step Solution

Answer

The completely factored form of the given expression is \((q-11)^2\).

Step 1: Identify the Form of the Perfect Square Trinomial

Observe that the expression $$q^2 - 22q + 121$$ resembles $$a^2 - 2ab + b^2$$, where \(a = q\). Our task is now to find \(b\).

Step 2: Find the Value of b

We can find the value of \(b\) using the coefficient of the \(q\) term in the expression. In our case, the coefficient is -22, and it represents -2ab. Since a = q, we have: $$-22 = -2(q)(b)$$ Divide both sides by -2q: $$\frac{-22}{-2q} = b$$ Now, insert the value of a: $$\frac{-22}{-2(q)}=b$$ $$\frac{-11}{q}=b$$ $$11=-b$$ So, \(b=11\).

Step 3: Write the Factored form of the Expression

Now that we have identified the values of \(a\) and \(b\), we can write the factored form of the given quadratic expression using the formula $$(a - b)^2$$. In our case, \(a=q\) and \(b=11\), so the expression can be written as: $$(q-11)^2$$ Therefore, the completely factored form of the given expression is $$(q-11)^2$$.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special form of a quadratic expression. It takes on the form: \(a^2 - 2ab + b^2\) or \(a^2 + 2ab + b^2\). This expression arises when you square a binomial.
In simpler terms, a binomial is an expression with two terms, like \((a+b)\) or \((a-b)\). When you square these, you get something like \((a-b)^2 = a^2 - 2ab + b^2\).
A primary identifier of a perfect square trinomial is that it represents a squared binomial in its expanded form.

• The first term is a perfect square (\(a^2\))
• The third term is also a perfect square (\(b^2\))
• The middle term is twice the product of the first and last term's square roots (\(-2ab\) or \(+2ab\))

Recognizing this form allows us to factor the trinomial quickly into a binomial squared, making it much easier to handle.

Quadratic Expressions

Quadratic expressions are polynomial expressions of the second degree. They typically look like \(ax^2 + bx + c\) where:

• \(a\), \(b\), and \(c\) are constants.
• \(x\) is a variable.
• \(a\) is not zero because it determines the quadratic nature.

Quadratics can be plotted as parabolas when graphed, and solving them typically finds the roots or zeros, the points where the parabola intersects the x-axis.
Quadratic expressions can come in various forms, including standard form, vertex form, and factored form. Understanding these forms allows you to solve quadratic equations and transform them to show different properties, such as roots or vertex location.

Factoring Techniques

Factoring is a technique used to break down more complex expressions into simpler ones, particularly quadratic expressions. Recognizing patterns in quadratics allows us to use different techniques to factorize them effectively.

• **Factoring by grouping**: Useful when there is no clear standard pattern, but terms can be grouped to expose a common factor.
• **Difference of squares**: Where expressions fit \(a^2 - b^2 = (a+b)(a-b)\).
• **Perfect square trinomials**: As discussed, these can be written as \((a-b)^2\) or \((a+b)^2\).
• **Quadratic formula**: Utilized when other methods don't immediately apply; expressed as \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\).
• **Trial and error**: Sometimes, testing simple combinations of factors leads to a solution.

Each technique has its use case depending on the structure of the given expression, and mastering them equips you to handle a variety of factoring challenges.