Problem 288 - Elementary Algebra

Question

In the following exercises, simplify. (a) \((3+\sqrt{5})^{2}\) (b) \((2-5 \sqrt{3})^{2}\)

Step-by-Step Solution

Answer

a) 14 + 6\text{ text } b) 202- 20 text .

Step 1: Identify the binomial form

Recognize that the expressions are binomials of the form .

Step 2: Expand using the binomial square formula

Use the formula to expand each term: For = For = 4.

Step 3: Simplify each expanded expression

Simplify the expanded expressions: = =.

Key Concepts

Binomial Expansion

Binomial expansion is a method used in algebra to expand expressions that are raised to a power. It is particularly useful when dealing with binomials, which are expressions containing two terms combined by addition or subtraction. For example, in the exercise provided, we have binomials such as \( (3 + \sqrt{5})^2 \) and \( (2 - 5 \sqrt{3})^2 \). When these expressions are squared, we need to use the expansion formula to simplify them.

Binomial expansion involves expanding the binomial using a specific formula, which transforms the expression into a polynomial. The formula for the square of a binomial \( (a + b)^2 \) is given by:

\[(a + b)^2 = a^2 + 2ab + b^2\]

This helps break down the expression, making it easier to simplify further. In the given steps, this formula was used to expand both examples.

Square of a Binomial

The square of a binomial is a special case of binomial expansion. It simplifies the process of squaring binomials by using the specific formula mentioned earlier. Let's look at two examples provided in the exercise:

* Example (a): \((3 + \sqrt{5})^2\)
* Example (b): \((2 - 5 \sqrt{3})^2\)

For example (a), using the formula \((a + b)^2 = a^2 + 2ab + b^2\), we identify \(a = 3\) and \(b = \sqrt{5}\). Plugging these into the formula gives:

\[ (3 + \sqrt{5})^2 = 3^2 + 2(3)(\sqrt{5}) + (\sqrt{5})^2\]

Simplifying further, we get:

\[ = 9 + 6\sqrt{5} + 5 = 14 + 6\sqrt{5}\]

For the second example, \((2 - 5 \sqrt{3})^2\), we have \(a = 2\) and \(b = -5 \sqrt{3}\). Substituting into the formula:

\[ (2 - 5 \sqrt{3})^2 = 2^2 + 2(2)(-5 \sqrt{3}) + (-5 \sqrt{3})^2\]

This expands to:

\[ = 4 - 20 \sqrt{3} + 75 = 79 - 20 \sqrt{3}\]

The process demonstrates the simplicity of using the square of a binomial formula to handle such problems.

Simplification of Expressions

After expanding the binomials, the next core step is the simplification of the resulting expressions. This process involves combining like terms and performing basic arithmetic operations to reach the simplest form.

In example (a):

\[ (3 + \sqrt{5})^2 = 9 + 6 \sqrt{5} + 5 = 14 + 6 \sqrt{5}\]

Here, we combined the constant terms \(9\) and \(5\) to get \(14\), while \(6 \sqrt{5}\) remains as it is since it's already in its simplest form.

Similarly, in example (b):

\[ (2 - 5 \sqrt{3})^2 = 4 - 20 \sqrt{3} + 75 = 79 - 20 \sqrt{3}\]

We combined \(4\) and \(75\) to get \(79\), while \(-20 \sqrt{3}\) remains unchanged as it's already simplified.

Simplification is crucial because it provides the final, most streamlined version of the expression, making it easier to understand and use in further calculations. Always take care to correctly identify and combine like terms for an accurate simplification.