Problem 90 - Algebra A Combined Function

Question

$$ \text { Multiply: }(\sqrt{5}-\sqrt{2}+1)^{2} $$

Step-by-Step Solution

Answer

The expanded expression is \(8 - 2\sqrt{10} + 2\sqrt{5} - 2\sqrt{2}\).

Step 1: Identify the Formula Needed

We need to expand \((\sqrt{5} - \sqrt{2} + 1)^2\). To do this, let's recognize it as an extension of the binomial square formula, \((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc\).

Step 2: Assign Variables

Assign \(a = \sqrt{5}\), \(b = -\sqrt{2}\), and \(c = 1\). Now we can expand using the formula \((a + b + c)^2\).

Step 3: Square Each Term

Calculate each square separately: \(a^2 = (\sqrt{5})^2 = 5\), \(b^2 = (-\sqrt{2})^2 = 2\), and \(c^2 = 1^2 = 1\).

Step 4: Calculate the Cross Terms

Calculate the cross-term products: - \(2ab = 2(\sqrt{5})(-\sqrt{2}) = -2\sqrt{10}\), - \(2ac = 2(\sqrt{5})(1) = 2\sqrt{5}\), - \(2bc = 2(-\sqrt{2})(1) = -2\sqrt{2}\).

Step 5: Sum All the Terms

Combine all terms from Step 3 and Step 4: \(a^2 + b^2 + c^2 + 2ab + 2ac + 2bc = 5 + 2 + 1 - 2\sqrt{10} + 2\sqrt{5} - 2\sqrt{2}\).

Step 6: Simplify the Expression

Add the constants and write the full expression:\(5 + 2 + 1 = 8\),So the expanded form is \(8 - 2\sqrt{10} + 2\sqrt{5} - 2\sqrt{2}\).

Key Concepts

Square of a Trinomial

Expanding expressions like \((a + b + c)^2\) can initially seem tricky, but it follows a structured process. When you square a trinomial, you extend the familiar formula for binomial squares. Instead of handling just two components, you're dealing with three, which results in more terms. The trinomial square formula is: \[(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc\]This formula can be broken down as follows:

• First, you square each individual term: \(a^2\), \(b^2\), and \(c^2\).
• Next, you consider all pairwise products of the terms and double them. This gives you \(2ab\), \(2ac\), and \(2bc\).
• Finally, you sum all of these results to get your expanded expression.

By systematically calculating each part, you ensure a complete and accurate expansion of the trinomial square.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. When working with expressions, it's crucial to understand each component:

• Numbers: Also known as constants, they have a fixed value.
• Variables: Symbols like \(x\), \(y\), or \(\sqrt{5}\) that can represent different values.
• Operations: Include addition, subtraction, multiplication, division, and powers.

In any algebraic work, like expanding a trinomial square, you're transforming an expression to another equivalent form. The goal might be to simplify, compute for specific inputs, or prepare it for a solution step. It's beneficial to recognize different expression forms and how they relate to each other.

Radicals in Algebra

Radicals involve expressions with roots, like square roots. They are a crucial part of algebra since they appear often in equations and expressions. The square root symbol \(\sqrt{}\) designates a number that, when multiplied by itself, yields the original number:

• For instance, \(\sqrt{5}\) indicates the number which, when squared, becomes 5.
• Radicals can also be negative, resulting from negative multipliers, such as \(-\sqrt{2}\) which still behaves as a typical square root in expressions.
• They follow specific rules when multiplied or simplified, such as \((\sqrt{a} \cdot \sqrt{b}) = \sqrt{ab}\).

It's essential to manage radicals carefully in algebra. They influence the solution process, particularly in operations like addition, multiplication, and rationalizing denominators. Understanding how to expand and simplify radicals is critical when dealing with algebraic expressions.