Problem 66 - Algebra A Combined Function

Question

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (\sqrt{y}-3 x)^{2} $$

Step-by-Step Solution

Answer

\(y - 6x\sqrt{y} + 9x^2\)

Step 1: Identify the expression

The expression given is \((\sqrt{y} - 3x)^2\). This expression is in the form of a binomial squared.

Step 2: Apply the Binomial Square Formula

Recall the binomial square formula: \((a - b)^2 = a^2 - 2ab + b^2\), where \(a = \sqrt{y}\) and \(b = 3x\).

Step 3: Square the first term

Square \(a\). Calculate \((\sqrt{y})^2\), which simplifies to \(y\).

Step 4: Calculate twice the product of the two terms

Calculate \(-2ab\). Substitute the values: \[ -2 \times \sqrt{y} \times 3x = -6x\sqrt{y} \]

Step 5: Square the second term

Square \(b\). Calculate \((3x)^2 = 9x^2\).

Step 6: Formulate the simplified expression

Combine all parts from steps 3, 4, and 5:\[ y - 6x\sqrt{y} + 9x^2 \]

Key Concepts

Binomial Theorem

The Binomial Theorem provides a way to expand expressions that are raised to a power. In our example, we have a binomial expression \((\sqrt{y} - 3x)^2\). Here, the expression consists of two terms: \(\sqrt{y}\) and \(3x\). The theorem is particularly useful for expressions like \((a+b)^n\), where either addition or subtraction is involved.

Using the theorem, these expressions can be expanded systematically. This is different from straightforward multiplication because it saves time and ensures accuracy by using a formula.

The binomial square formula, a specific case of the Binomial Theorem for \((a-b)^2\), is given by:

• \((a-b)^2 = a^2 - 2ab + b^2\)

It allows us to quickly determine the squared expression of our binomial by clarifying what each term in the expansion will contribute.

Square of a Binomial

Squaring a binomial means multiplying it by itself, like in our expression \((\sqrt{y} - 3x)^2\). The formula applies here, determining how each part of the binomial contributes to the expansion. Let's break this down:

1. **First Term Squared:** This is calculated by taking the first term, \(\sqrt{y}\), and squaring it, resulting in \(y\).
2. **Twice the Product of Both Terms:** Next, we multiply the terms together and then double it: \(-2 \times \sqrt{y} \times 3x = -6x\sqrt{y}\).
3. **Square of the Second Term:** This is done by squaring the second term, \(3x\), which results in \(9x^2\).

Combining these parts gives us the expanded form \(y - 6x\sqrt{y} + 9x^2\). This formula allows the binomial to unfold neatly into a quadratic expression, making it easier to work with.

Simplifying Expressions

Once we expand expressions like \((\sqrt{y} - 3x)^2\) using the binomial square formula, the next step is simplification. Simplifying simplifies expressions to their most straightforward form, making calculations easier and more intuitive.

Key simplifying techniques include:

• **Combining Like Terms:** Here, there are no like terms to combine, but this process helps reduce the expression's complexity when similar terms are involved.
• **Factoring or Distributing as Needed:** Occasionally, redistributing terms can further simplify, though in our case the expression \(y - 6x\sqrt{y} + 9x^2\) is already simplified.

Simplification not only assists in making equations manageable but also helps us see the resultant relationships between variables more clearly. It turns complex expressions into clear, concise, and easily interpreted equations.