Problem 36 - Algebra and Trigonometry

Question

31–76 ? Factor the expression completely. $$ x^{2}-14 x+48 $$

Step-by-Step Solution

Answer

The expression factors to \((x - 6)(x - 8)\).

Step 1: Understanding the Problem

The problem asks us to factor the quadratic expression \( x^2 - 14x + 48 \) completely. Factoring a quadratic involves expressing it as a product of two binomials.

Step 2: Identify the Form

The given expression is in the standard quadratic form \( ax^2 + bx + c \), where \( a = 1 \), \( b = -14 \), and \( c = 48 \).

Step 3: Finding Two Numbers

We need two numbers that multiply to \( c = 48 \) and add up to \( b = -14 \). These numbers are \(-6\) and \(-8\).

Step 4: Write as a Product of Two Binomials

With the numbers found, the quadratic expression \( x^2 - 14x + 48 \) can be factored as \( (x - 6)(x - 8) \).

Step 5: Verify the Factorization

Expand \( (x - 6)(x - 8) \) to ensure it equals the original expression:\[(x - 6)(x - 8) = x^2 - 8x - 6x + 48 = x^2 - 14x + 48\]This confirms the factorization is correct.

Key Concepts

Quadratic Expressions

A quadratic expression is a polynomial of degree two, often appearing in the form \( ax^2 + bx + c \). The most noticeable feature of quadratic expressions is the presence of the squared term, which determines its degree. Here, \( a, b, \) and \( c \) are constants, with \( a \) not equal to zero since this ensures the expression remains quadratic rather than linear or constant. Quadratics can often be challenging because of their various forms and solutions, but they are fundamental in algebra due to their wide applications in areas such as physics, engineering, and economics. The expression \( x^2 - 14x + 48 \) is such an example, where the coefficients \( a = 1 \), \( b = -14 \), and \( c = 48 \) set its characteristics. Understanding how to manipulate and solve quadratic expressions is a cornerstone of algebraic proficiency.

Binomial Factorization

Factoring is the process of breaking down an equation or expression into simpler parts—or factors—that when multiplied together give the original expression. In binomial factorization, we specifically look to break down quadratic expressions into two simpler binomial expressions.
For the quadratic \( x^2 - 14x + 48 \), we find two numbers that add up to \(-14\) (the coefficient of \(x\)) and multiply to \(48\) (the constant term). These numbers are \(-6\) and \(-8\). This method takes advantage of the product-sum relationship, common to quadratics:

• The two numbers should multiply to the constant term \(c\).
• They should add up to the linear coefficient \(b\).

When successful, this allows us to rewrite the quadratic \( x^2 - 14x + 48 \) as \( (x - 6)(x - 8) \). This form is much simpler to work with and understand, particularly when solving quadratic equations or interpreting graphs.

Standard Quadratic Form

A quadratic equation in standard form is written as \( ax^2 + bx + c = 0 \). Understanding this standard form is crucial because it provides a consistent structure for various factorization and solution techniques. In our example, the quadratic expression \( x^2 - 14x + 48 \) is already in standard form where \( a = 1 \), \( b = -14 \), and \( c = 48 \).
Recognizing this form is important for several reasons:

• It establishes the framework for using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• It helps in identifying approaches like completing the square or graphing.
• It enables straightforward factorization into binomial pairs for simpler expressions.

Mastery of the standard quadratic form makes solving and interpreting quadratics less daunting, as it becomes a matter of methodically applying consistent strategies.