Problem 35 - Algebra and Trigonometry

Question

31–76 ? Factor the expression completely. $$ x^{2}-2 x-8 $$

Step-by-Step Solution

Answer

The factored form is \((x - 4)(x + 2)\).

Step 1: Identify the Coefficients

To factor the quadratic expression \( x^2 - 2x - 8 \), begin by identifying the coefficients: \( a = 1 \), \( b = -2 \), and \( c = -8 \).

Step 2: Find Two Numbers that Multiply and Add

Look for two numbers whose product is \( a \cdot c = 1 \cdot (-8) = -8 \) and whose sum is \( b = -2 \). These numbers are \( -4 \) and \( 2 \) because \( -4 \times 2 = -8 \) and \( -4 + 2 = -2 \).

Step 3: Rewrite the Middle Term

Rewrite the expression \( x^2 - 2x - 8 \) by splitting the middle term using the numbers found: \( x^2 - 4x + 2x - 8 \).

Step 4: Factor by Grouping

Group the terms: \( (x^2 - 4x) + (2x - 8) \). Factor out the greatest common factor from each group: \( x(x - 4) + 2(x - 4) \).

Step 5: Factor Out the Common Binomial

The expression has a common binomial \( x - 4 \). Factor it out to get: \( (x - 4)(x + 2) \).

Key Concepts

Quadratic Equations

Quadratic equations are a special type of polynomial equation. These equations are characterized by having a degree of two, which means their highest power is squared or raised to the power of two. The general form of a quadratic equation is \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients.
Understanding quadratic equations is essential because they appear frequently in algebra and have various applications in fields such as physics, engineering, and finance. They can describe parabolic motion, structural curves, and growth phenomena.
When solving quadratic equations, there are multiple methods available:

• Factoring, which is breaking it down into simpler expressions.
• Using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \).
• Completing the square, which involves rewriting the equation so one side forms a perfect square trinomial.

These techniques allow us to find the roots or solutions of the quadratic equation, which are the values of \( x \) that satisfy the equation.

Factoring by Grouping

Factoring by grouping is a method used to simplify algebraic expressions. This technique is especially helpful when dealing with quadratic expressions or equations that don't directly factor into a product of binomials.
When factoring by grouping, you split the expression into pairs or groups that have a common factor. For example, consider the expression \( x^2 - 2x - 8 \). Through factoring by grouping, this expression is rewritten with its middle term split into two terms, resulting in \( x^2 - 4x + 2x - 8 \).
To proceed with factoring by grouping:

• Identify pairs of terms in the expression: \((x^2 - 4x) + (2x - 8)\).
• Extract a common factor from each group: \(x(x - 4) + 2(x - 4)\).
• Notice the common binomial \(x - 4\), and factor it out: \((x - 4)(x + 2)\).

This method can simplify more complex algebraic expressions, making them easier to solve or analyze.

Algebraic Expressions

An algebraic expression is a combination of variables, numbers, and operations (such as addition, subtraction, multiplication, and division). These expressions do not contain an equality sign, which makes them different from algebraic equations.
Algebraic expressions can range from simple monomials like \( 3x \) to more complex polynomials like \( x^2 - 2x - 8 \). Understanding these expressions is fundamental in algebra, as they represent real-world quantities and relationships.
Key elements to consider in algebraic expressions include:

• Variables: Symbols such as \( x \), \( y \), or \( z \) that represent unknown values.
• Coefficients: Numbers that multiply the variables, like the \( 1 \) in \( 1x^2 \).
• Terms: Parts of the expression separated by addition or subtraction, like \( x^2 \), \(-2x\), and \(-8\) in the expression given.

By learning to manipulate algebraic expressions, students can solve equations, simplify expressions, and tackle a wide variety of math problems. These skills are crucial for higher-level math and many practical applications.