Problem 97 - Introductory Algebra for College Students

Question

Explain how to factor \(2 x^{2}-x-1\).

Step-by-Step Solution

Answer

The factored form of the equation \(2 x^{2}-x-1\) is \( (2x + 1)(x - 1) \).

Step 1: Identify the coefficients

From the equation \(2 x^{2}-x-1\), the coefficients a, b, and c are 2, -1, and -1 respectively.

Step 2: Multiply a and c

Multiply the coefficients a and c. Here, \( ac = 2 * -1 = -2 \). Now we need to find two numbers that add up to -1 (b) and multiply to -2 (ac).

Step 3: Find numbers

The numbers -2 and 1 add up to -1 and multiply to -2. Therefore, these are the numbers we're looking for.

Step 4: Break up the middle term

Break up the middle term of the original equation using the numbers found. This gives us \(2x^2 -2x +x -1\).

Step 5: Factor by grouping

Group the terms to factor by grouping: \((2x^{2} - 2x) + (x - 1) \). This simplifies to \(2x(x - 1) + 1(x - 1)\).

Step 6: Extract common factor

Now we see that \( (x - 1) \) is a common factor. Extracting this gives us the factored equation: \( (2x + 1)(x - 1) \).

Key Concepts

Coefficients Identification

Identifying the coefficients in a quadratic expression is one of the first and fundamental steps in factoring. A standard form of a quadratic equation is given by \( ax^2 + bx + c \).
Here, \( a \), \( b \), and \( c \) are the coefficients, with \( a \) being the coefficient of \( x^2 \), \( b \) the coefficient of \( x \), and \( c \) the constant term.For the expression \( 2x^2 - x - 1 \), identify the coefficients:

• \( a = 2 \)
• \( b = -1 \)
• \( c = -1 \)

Accurate identification is important because it sets the stage for subsequent steps in the factoring process.
Make sure to include the signs (positive or negative) of the coefficients as they will influence the calculations.

Factoring by Grouping

Factoring by grouping is a valuable technique used to factor quadratic expressions. It involves breaking an expression into groups, which are simpler to factor individually. The aim is to find a common factor in each group, and then factor it out.Let's go through this using \( 2x^2 - x - 1 \).
First, we break up the middle term \( -x \) using the numbers found that add to \( b \) and multiply to \( ac \). In this case, we use \( -2x \) and \( x \) because \( -2 + 1 = -1 \).
The expression becomes \( 2x^2 - 2x + x - 1 \).
Now, group terms: \((2x^2 - 2x) + (x - 1) \).Factor out the common factor from each group:

• From \( 2x^2 - 2x \), factor out \( 2x \), resulting in \( 2x(x - 1) \).
• From \( x - 1 \), factor out \( 1 \), resulting in \( 1(x - 1) \).

This technique simplifies expressions and facilitates further factoring steps.

Quadratic Expression

A quadratic expression is any expression that can be reduced to a form involving a term with \( x^2 \). It usually takes the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. Quadratic expressions often arise in various applications, from physics to economics.In the context of factoring, understanding the structure of a quadratic expression helps us apply appropriate techniques to simplify it into the form of \((px + q)(rx + s)\).
This makes solving equations easier. Recognizing whether an expression is quadratic is the first step in determining the right approach for factoring.By breaking down the expression into simpler components, we can focus on underlying patterns that allow for factoring to be performed accurately and efficiently.

Factoring Techniques

Factoring techniques refer to different strategies used to break down a complicated expression into simpler, multiplicable components. There are various techniques, but choosing the most appropriate one depends largely on the structure of the expression.In factoring \( 2x^2 - x - 1 \), we used:

• **The AC Method**: We multiplied the coefficients \( a \) and \( c \), and found numbers that add to \( b \) and multiply to \( ac \).
• **Factoring by Grouping**: As covered, this involved creating groups within the expression that could yield common factors.

Both methods emphasize the importance of manipulating coefficients correctly. Knowing and applying these techniques allow for efficient simplification and solving of expressions.
Practicing different techniques enhances problem-solving skills and mathematical dexterity, essential for handling diverse quadratic expressions.