Problem 66 - Intermediate Algebra

Question

Multiply and then simplify if possible. $$ (\sqrt{y}-3 x)^{2} $$

Step-by-Step Solution

Answer

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

Step 1: Recognize the Expression

The expression given is \((\sqrt{y} - 3x)^2\). This is a binomial expression raised to the power of 2, which means we will apply the formula for the square of a binomial, \((a - b)^2 = a^2 - 2ab + b^2\).

Step 2: Identify Variables in the Formula

In the binomial \((\sqrt{y} - 3x)\), the terms are identified as follows: \(a = \sqrt{y}\) and \(b = 3x\).

Step 3: Apply the Binomial Square Formula

Apply the formula: \((\sqrt{y})^2 - 2(\sqrt{y})(3x) + (3x)^2\). This results in \(y - 6x\sqrt{y} + 9x^2\).

Step 4: Simplify the Expression

The expression \(y - 6x\sqrt{y} + 9x^2\) is in its simplest form as all the terms are already simplified.

Key Concepts

Simplifying Expressions

To fully understand the process of simplifying expressions, let's first define what it means. Simplifying an expression involves reducing it to its most straightforward form without changing its value. This often includes eliminating parentheses and combining like terms.
To illustrate, consider an expression like \(a^2 + 2ab + b^2\). Once simplified, it might become a simpler polynomial or even a single term, given certain conditions.

• Look for common terms that can be combined.
• Remove parentheses by distributing and combining like terms.
• Ensure all terms are in their simplest forms, including roots and exponents.

In the expression \((\sqrt{y} - 3x)^2\), simplifying means expanding and reducing it into a format where it can be more easily interpreted and used in further calculations or problem-solving scenarios.

Algebraic Multiplication

Algebraic multiplication involves distributing terms over additions or subtractions within expressions. This means each term of one expression is multiplied by each term in another expression.

• For a term like \((a-b)^2\), multiplication involves: * Squaring the first term: \(a^2\) * Multiplying each term by each other and doubling: \(-2ab\) * Squaring the second term: \(b^2\)

In our exercise of expanding \((\sqrt{y}-3x)^2\), we apply the multiplication across terms: first multiply \(\sqrt{y}\) with itself, then \(\sqrt{y}\) with \(3x\), and \(3x\) with itself.
This highlights the distributive property, where every part of the expression interacts with all others through multiplication.

Square of a Binomial

The square of a binomial is a valuable algebraic identity used in expanding expressions that have the form \((a \pm b)^2\). This formula is

• \((a + b)^2 = a^2 + 2ab + b^2\)
• \((a - b)^2 = a^2 - 2ab + b^2\)

By using this identity, you can quickly expand binomials without manually multiplying each term.
In our given expression \((\sqrt{y} - 3x)^2\),

• where \(a = \sqrt{y}\) and \(b = 3x\), using the identity \((a - b)^2\), results in \(a^2 - 2ab + b^2\), or \(y - 6x\sqrt{y} + 9x^2\).

This process simplifies the complex-looking expression into a more user-friendly form, which is easier to work with in additional algebraic operations.