Problem 42 - Elementary Algebra

Question

For the following problems, find the products. $$ (x+5.2)^{2} $$

Step-by-Step Solution

Answer

Answer: The simplified product of the given expression is \(x^2 + 10.4x + 27.04\).

Step 1: Write down the given expression

We are given the following expression: $$(x+5.2)^{2}$$

Step 2: Expand the expression using distributive property

To expand the expression, rewrite \((x+5.2)^{2}\) as \((x+5.2)(x+5.2)\) and then use the distributive property to combine the terms. $$(x+5.2)(x+5.2) = x(x+5.2) + 5.2(x+5.2)$$

Step 3: Distribute each term

Distribute each term from the expression in step 2. $$x(x+5.2) + 5.2(x+5.2) = x^2 + 5.2x + 5.2x + 27.04$$

Step 4: Combine like terms

Combine the like terms (5.2x and 5.2x) in the expression. $$x^2 + 5.2x + 5.2x + 27.04 = x^2 + 10.4x + 27.04$$

Step 5: Write the final result

The simplified product of \((x+5.2)^{2}\) is given by: $$(x+5.2)^{2} = x^2 + 10.4x + 27.04$$

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps us simplify expressions. When dealing with multiplication over addition, the distributive property allows us to multiply each term inside a set of parentheses by a number or another expression outside the parentheses. In more formal terms, for any numbers or expressions \(a\), \(b\), and \(c\), the property can be stated as:

• \(a(b+c) = ab + ac\)

In the context of our exercise, we start with \((x+5.2)^2\), which is rewritten as \((x+5.2)(x+5.2)\). By using the distributive property, we multiply \(x\) with every term in the second \((x+5.2)\), and then multiply \(5.2\) with every term in the same second \((x+5.2)\). This results in:

• \(x(x+5.2) + 5.2(x+5.2)\)
• Which further simplifies to: \(x^2 + 5.2x + 5.2x + 27.04\)

This step lays the foundation for us to further simplify the expression by combining like terms.

Expansion

Expansion involves rewriting an expression in an extended form using multiplication. It is a crucial step in simplifying or solving algebraic problems. In our exercise, the expansion process takes a single term expression like \((x+5.2)^2\) and converts it to \((x+5.2)(x+5.2)\).

When expanding, each component inside the first set of parentheses is doubled by the components in the second set of parentheses. This leads to the increased expression:

• First term: \(x(x+5.2)\)
• Second term: \(5.2(x+5.2)\)

The resulting expression \(x^2 + 5.2x + 5.2x + 27.04\) clearly displays the extended version of the initial squared expression.

Through expansion, the expression becomes easier to handle with multiple like terms now present, which will be combined in the next algebraic step.

Combining Like Terms

Combining like terms is the process of simplifying an expression by adding or subtracting terms that have the same variables raised to the same power. This step helps to reduce the number of terms in an expansion, making the expression more manageable and easier to interpret.

In the case of our exercise, we deal with the expression \(x^2 + 5.2x + 5.2x + 27.04\). Here, the terms with \(x\) are considered 'like terms' because they both have \(x\) raised to the power of 1. By adding them together, we achieve:

• \(5.2x + 5.2x = 10.4x\)

Combining like terms simplifies the expression further into \(x^2 + 10.4x + 27.04\).

Such simplification is crucial in algebra as it allows for clearer understanding and easier manipulation of expressions, allowing students to focus more on solving and interpreting complex algebraic problems.