Problem 38 - Elementary Algebra

Question

Factor. $$ a_{2}-22 a+120 $$

Step-by-Step Solution

Answer

The factored form is \((a - 10)(a - 12)\).

Step 1: Identify Coefficients

The quadratic expression given is \(a^2 - 22a + 120\). Here, the coefficient of \(a^2\) is 1, the coefficient of \(a\) is -22, and the constant term is 120.

Step 2: Set up the Factorization Expression

We are looking to factor the expression into the form \((a + m)(a + n)\) such that \(m + n = -22\) and \(m \cdot n = 120\).

Step 3: Find the Pair of Numbers

To find \(m\) and \(n\), we look for factors of 120 that add up to -22. These numbers are -10 and -12 because \(-10 + (-12) = -22\) and \(-10 \times -12 = 120\).

Step 4: Write the Factored Form

Using the values of \(m\) and \(n\) found in Step 3, the expression \(a^2 - 22a + 120\) factors as \((a - 10)(a - 12)\).

Key Concepts

Quadratic Expression

A quadratic expression is a type of polynomial, specifically a binomial, that involves the square of the variable. It generally takes the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. This type of expression is significant because it appears often in algebra, especially in relation to parabolas, projectile motion, and even in areas like business and engineering.

In the case of the expression given, \(a^2 - 22a + 120\), it is organized as a quadratic expression where the variable \(a\) is squared. The quadratic expression is complete when it involves all three terms: a term with the square of the variable, a linear term, and a constant. Recognizing this structure helps students understand the roles of each component in the expression, guiding them in operations like factoring.

Coefficients

Coefficients are the numerical factors present in terms of an expression. In a quadratic expression like \(ax^2 + bx + c\), the coefficient of \(x^2\) is \(a\), the coefficient of \(x\) is \(b\), and \(c\) is the constant term—not technically a coefficient but plays a similar role.

For the expression \(a^2 - 22a + 120\), the coefficients are as follows:

• The coefficient of \(a^2\) is 1. This is an implied "1" that controls the quadratic term's degree.
• The coefficient of \(a\) is -22, guiding the linear term in how much it ought to increase or decrease.
• The constant term is 120, which shifts the entire graph vertically and is pivotal to finding roots.

Understanding coefficients helps in various algebraic operations, such as solving equations or in this case, factoring.

Factorization

Factorization involves breaking down an expression into products of simpler expressions, essentially reversing expansion. It helps solve quadratic equations by making it easier to find the roots or solutions.

The factorization of a quadratic expression like \(a^2 - 22a + 120\) involves expressing it as a product of two binomials. For this expression, it means finding two numbers (let's call them \(m\) and \(n\)) that satisfy:

• The sum \(m + n = -22\)
• The product \(m \cdot n = 120\)

Here, the numbers are -10 and -12 since they meet both conditions. This results in the factored form \((a - 10)(a - 12)\). This process simplifies the quadratic into linear factors, allowing easier solving or analysis. Knowing how to factor quadratics yields several applications, including solving real-world problems modeled by quadratic functions.