Simplifying Algebraic Expressions: Expanding and Combining Terms

The question asks us to simplify the algebraic expression \( (3x + 5)^2 + (3x - 5)^2 \). This involves expanding the squared binomials and then combining the resulting terms.

Understanding Binomial Expansion

We use standard algebraic identities to expand binomial squares:

• The square of a sum: \( (a + b)^2 = a^2 + 2ab + b^2 \)
• The square of a difference: \( (a - b)^2 = a^2 - 2ab + b^2 \)

Let's apply these identities to the given expression.

Step-by-Step Simplification

Step 1: Expand the first term \( (3x + 5)^2 \)

Here, \( a = 3x \) and \( b = 5 \). Using the identity \( (a + b)^2 = a^2 + 2ab + b^2 \):

\( (3x + 5)^2 = (3x)^2 + 2(3x)(5) + (5)^2 \)

Calculate each part:

• \( (3x)^2 = 3^2 \times x^2 = 9x^2 \)
• \( 2(3x)(5) = 2 \times 3 \times x \times 5 = 30x \)
• \( (5)^2 = 25 \)

So, \( (3x + 5)^2 = 9x^2 + 30x + 25 \).

Step 2: Expand the second term \( (3x - 5)^2 \)

Here, \( a = 3x \) and \( b = 5 \). Using the identity \( (a - b)^2 = a^2 - 2ab + b^2 \):

\( (3x - 5)^2 = (3x)^2 - 2(3x)(5) + (5)^2 \)

Calculate each part:

• \( (3x)^2 = 9x^2 \)
• \( -2(3x)(5) = -2 \times 3 \times x \times 5 = -30x \)
• \( (5)^2 = 25 \)

So, \( (3x - 5)^2 = 9x^2 - 30x + 25 \).

Step 3: Add the expanded terms

Now, we add the results from Step 1 and Step 2:

\( (3x + 5)^2 + (3x - 5)^2 = (9x^2 + 30x + 25) + (9x^2 - 30x + 25) \)

Combine like terms:

• Combine the \( x^2 \) terms: \( 9x^2 + 9x^2 = 18x^2 \)
• Combine the \( x \) terms: \( 30x - 30x = 0x = 0 \)
• Combine the constant terms: \( 25 + 25 = 50 \)

So, the simplified expression is \( 18x^2 + 0 + 50 = 18x^2 + 50 \).

Step 4: Factor the simplified expression

The expression is \( 18x^2 + 50 \). We can factor out the greatest common divisor of 18 and 50, which is 2.

\( 18x^2 + 50 = 2(9x^2) + 2(25) = 2(9x^2 + 25) \)

The simplified form of the expression \( (3x + 5)^2 + (3x - 5)^2 \) is \( 2(9x^2 + 25) \).

Summary of Steps

Here's a quick look at the steps:

1. Expand the first binomial using \( (a+b)^2 \) formula.
2. Expand the second binomial using \( (a-b)^2 \) formula.
3. Add the results from step 1 and step 2.
4. Combine like terms.
5. Factor the final expression if possible.

Comparing with Options

Let's compare our simplified expression \( 2(9x^2 + 25) \) with the given options:

• Option 1: \( 450x \) - Does not match.
• Option 2: \( 500x \) - Does not match.
• Option 3: \( 9x^2 + 50 \) - Does not match \( 18x^2 + 50 \).
• Option 4: \( 2(9x^2 + 25) \) - Matches our result \( 2(9x^2 + 25) \).

Revision Table: Algebraic Identities

Additional Information: Why Simplification is Important

Simplifying algebraic expressions is a fundamental skill in mathematics. It helps in:

• Making expressions easier to understand and work with.
• Solving equations more efficiently.
• Identifying equivalent expressions.
• Preparing expressions for graphing or further calculations.

In this problem, simplifying \( (3x + 5)^2 + (3x - 5)^2 \) from a longer form to \( 2(9x^2 + 25) \) makes the structure clearer and potentially easier to evaluate for specific values of \( x \).