Problem 95 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ (6 x-4)^{2} $$

Step-by-Step Solution

Answer

Answer: The simplified expression for \((6x-4)^2\) is \(36x^2 - 48x + 16\).

Step 1: Rewrite the expression as a product

The given expression is \((6x-4)^2\). We can rewrite this as the product of the expression with itself: \((6x-4)(6x-4)\).

Step 2: Apply the distributive property

Now, we will apply the distributive property by multiplying each term in the first parentheses by each term in the second parentheses. \((6x-4)(6x-4) = (6x)(6x) - (6x)(4) - (4)(6x) + (4)(4)\)

Step 3: Simplify terms

We will now simplify each term obtained in the previous step. \((6x)(6x) = 36x^2\) \(- (6x)(4) = - 24x\) \(- (4)(6x) = - 24x\) \((4)(4) = 16\)

Step 4: Combine like terms

Lastly, we will combine the like terms in the expression to obtain our simplified expression. \(36x^2 - 24x - 24x + 16 = 36x^2 - 48x + 16\) The simplified expression for \((6x-4)^2\) is \(36x^2 - 48x + 16\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps in expanding expressions. This property allows us to multiply a single term by two or more terms within a set of parentheses. It can be expressed in general form as:

• For any numbers or variables, \(a(b + c) = ab + ac\).


• Similarly, \(a(b - c) = ab - ac\).

In this context, to simplify the expression \((6x-4)^2\), we first rewrite it as \((6x-4)(6x-4)\). Using the distributive property, each term in the first set of parentheses is multiplied by each term in the second set of parentheses. This means:

• \((6x)(6x)\)


• \((6x)(-4)\)


• \((-4)(6x)\)


• \((4)(-4)\)

This approach breaks down the multiplication process into manageable steps and sets the stage for further simplification.

Like Terms

Like terms are terms in an algebraic expression that have the same variables raised to the same power. They are essential when simplifying expressions because like terms can be combined through addition or subtraction. For example:

• In the expression \(36x^2 - 24x - 24x + 16\), the terms \(-24x\) and \(-24x\) are like terms because they both consist of the variable \(x\) raised to the first power.

Unlike terms such as \(36x^2\), which has \(x^2\), cannot be combined with terms that have \(x\) raised to a different power.
To simplify further, we add or subtract the coefficients of the like terms. In our example, this results in:

• Combining \(-24x - 24x\) gives \(-48x\).

Recognizing and combining like terms is crucial for simplifying algebraic expressions efficiently.

Simplification

Simplification in algebra involves reducing an expression to its simplest form. This often includes using properties like the distributive property and combining like terms. After applying these techniques to our exercise step by step, the expression is greatly simplified.Let's examine the simplification process for our expression \((6x-4)^2\):

• We began by multiplying the expression with itself, using the distributive property: \(36x^2 - 24x - 24x + 16\).
• Next, we identified and combined the like terms: \(-24x - 24x = -48x\).
• The final simplified form of the expression is \(36x^2 - 48x + 16\).

The goal of simplification is to produce an expression that is easy to read and work with, which often involves shortening it while retaining the same value. Through careful calculation and understanding of properties, complex expressions can be effectively simplified.