Problem 152 - Introductory Algebra for College Students

Question

Exercises 150–152 will help you prepare for the material covered in the next section. Factor: \((x-2)(x+3)-6\)

Step-by-Step Solution

Answer

The factored form of the expression \((x-2)(x+3)-6\) is \((x - 4)(x + 3)\).

Step 1: Expand the brackets

First, let's expand the binomial expressions in the brackets. This is done by distributing \(x\) in \((x-2)\) and \((x+3)\), leading to the new expression: \(x^2 + 3x - 2x - 6 - 6\).

Step 2: Simplify.

Now, let's simplify by combining like terms. The \(+3x\) and \(-2x\) combine to yield \(x\), and the \(-6 -6\) combine to give \(-12\). So now we have: \(x^2 + x - 12\).

Step 3: Factor the quadratic expression.

To factor the quadratic expression \(x^2 + x - 12\), we need to find two numbers that add up to 1 (the coefficient of \(x\)) and multiply to -12 (the independent term). Those numbers are 4 and -3. We can now rewrite the quadratic expression as \((x - 4)(x + 3)\).

Key Concepts

Quadratic Expressions

Understanding quadratic expressions is crucial for solving a variety of algebra problems. Quadratic expressions are polynomials that take the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\).
These expressions can be factored, expanded, or analyzed to find roots, which are the values of \(x\) that make the expression zero.

• Familiar Forms: The standard form is \(ax^2 + bx + c\), whereas the factored form usually looks like \((dx + e)(fx + g)\).
• Roots and Solutions: In a quadratic expression, the solutions are the values of \(x\) which satisfy the equation \(ax^2 + bx + c = 0\).

Grasping these forms helps to manipulate and solve these types of equations efficiently. In many algebraic operations, especially in factoring and solving these expressions, the relationship between numbers, such as their sum and product, comes into play, as it helps to quickly factor or solve the equations.

Binomial Expansion

Binomial expansion is a method used to expand expressions that are a product of two binomials, such as \((x - 2)(x + 3)\) from our original exercise.
It involves the application of the distributive property repeatedly to each term inside the brackets.

• The Process: First, multiply each term in the first binomial by each term in the second binomial.
• Combining Like Terms: After multiplying, combine like terms to simplify the expression.

For example, expanding \((x - 2)(x + 3)\) gives us \(x^2 + 3x - 2x - 6\).
Further, combining like terms yields \(x^2 + x - 6\). Practicing binomial expansion helps simplify expressions into more manageable forms and is foundational for algebraic manipulation.

Simplifying Expressions

Simplifying expressions is a key skill in algebra that makes complex equations easier to work with. Once you've expanded an expression, the next step is to combine like terms.
Like terms are terms that have the same variable raised to the same power.

• Identification: Look for numbers with the same variables and exponents to combine. For example, in \(3x - 2x\), both terms have \(x\) as a variable, so they are combined.
• Combining: Add or subtract their coefficients, as shown in the expression \(3x - 2x = x\).

By simplifying terms such as \(-6 - 6\) to \(-12\), we reduce the complexity of the expression.
This step helps not only in solving but also in factoring the expressions into components, allowing for deeper analysis and solution finding.