Problem 40 - Algebra A Combined Function

Question

Factor each completely. $$ 4 x^{2}-36 $$

Step-by-Step Solution

Answer

The fully factored form is \(4(x - 3)(x + 3)\).

Step 1: Identify the common factor

Look at each term in the expression \(4x^2 - 36\). Notice that both terms can be divided by the common factor of 4. Factoring out the common factor 4, we get \(4(x^2 - 9)\).

Step 2: Recognize the difference of squares

The expression inside the parentheses \((x^2 - 9)\) is a difference of squares. Recall that a difference of squares can be factored as \(a^2 - b^2 = (a - b)(a + b)\). Here \(x^2 - 9\) fits this pattern with \(a = x\) and \(b = 3\).

Step 3: Apply the difference of squares formula

Using the formula from the previous step, \(x^2 - 9\) can be factored into \((x - 3)(x + 3)\).

Step 4: Write the fully factored form

Substitute the factored result from Step 3 back into the expression from Step 1. Thus, \(4(x^2 - 9)\) fully factors to \(4(x - 3)(x + 3)\).

Key Concepts

Common Factor

In mathematics, identifying a common factor is a helpful strategy when factoring quadratic expressions or any algebraic expressions. The common factor refers to a number or expression that divides each term in the expression without leaving a remainder.
For example, consider the quadratic expression given in the exercise, \(4x^2 - 36\). Both terms, \(4x^2\) and \(36\), can be divided by 4. Here, 4 is the common factor.
When we factor out the common factor from each term, we simplify the expression as follows:

• Divide each term in the expression by the common factor (4):
• \(4x^2/4 = x^2\)
• \(-36/4 = -9\)

After factoring out the common factor 4, the simplified expression becomes \(4(x^2 - 9)\). By extracting the common factor first, we make further factoring processes easier to manage.

Difference of Squares

A difference of squares is a special algebraic expression that takes the form \(a^2 - b^2\). It is called a 'difference' because it involves a subtraction, and 'squares' because both terms are perfect squares.
Recognizing a difference of squares is key to factoring it. The important formula to remember is:

• \(a^2 - b^2 = (a - b)(a + b)\)

In our exercise, after factoring out the common factor, the expression inside the parentheses is \(x^2 - 9\). Here, \(x^2\) is the square of \(x\), and \(9\) is the square of \(3\), meaning \(a = x\) and \(b = 3\).
Thus, using our difference of squares formula:

• \(x^2 - 9 = (x - 3)(x + 3)\)

By learning this formula, you can quickly factor any expression that is a difference of squares.

Factoring Process

The factoring process is a series of steps to break down expressions into simpler and more useful forms. This process is particularly important when solving equations, as it helps identify the roots or solutions.In the given exercise, the factoring process involves:

• Step 1: Identifying and factoring out any common factors first. For \(4x^2 - 36\), the common factor is 4, resulting in \(4(x^2 - 9)\).
• Step 2: Recognizing any special patterns, such as the difference of squares, within the factored expression. The \(x^2 - 9\) is a difference of squares.
• Step 3: Utilizing known formulas to factor the remaining parts of the expression. Applying the formula \(a^2 - b^2 = (a - b)(a + b)\) gives \((x - 3)(x + 3)\).
• Step 4: Writing the fully factored form. Substitute the result from the special pattern formula back into the expression to get \(4(x - 3)(x + 3)\).

Following these steps will help ensure complete and correct factorization of quadratic expressions.