Problem 40 - Algebra 1: Concepts and Skills

Question

Find the product. $$ (4 n-3)^{2} $$

Step-by-Step Solution

Answer

\((4n - 3)^2 = 16n^2 - 24n + 9\

Step 1: Apply the binomial formula

Apply formula \((a - b)^2 = a^2 - 2ab + b^2\) to \((4n - 3)^2\). Here \(a=4n\) and \(b=3\). Thus we get \((4n)^2 - 2*(4n)*3 + 3^2\).

Step 2: Compute the squares

Calculate \(a^2\) and \(b^2\), i.e., \((4n)^2\) and \(3^2\). We get \(16n^2\) and \(9\) respectively. Replace these values in the result from step 1 to yield \(16n^2 - 2*(4n)*3 + 9\).

Step 3: Do the multiplication

Now calculate \(2 * 4n * 3\). We get \(24n\). Replace this value in the result from step 2 to get \(16n^2 - 24n + 9\).

Step 4: Final Simplification

The expression \(16n^2 - 24n + 9\) is already in its simplest form, and is thus the final solution.

Key Concepts

Polynomial Expansion

Expanding a polynomial involves breaking down an expression like \((a - b)^2\) into simpler parts. This process is often accomplished using the binomial theorem. In our original exercise, we are tasked with expanding \((4n - 3)^2\).

The binomial theorem states that for any binomial \(a + b\), raised to the power of 2, you can use the formula:

• \((a - b)^2 = a^2 - 2ab + b^2\)

This formula helps in transforming the polynomial into an expanded form. It simplifies multiplication and makes it easier to solve complex algebraic expressions.

In the case of our problem, the expansion allows us to rewrite \((4n - 3)^2\) as \(16n^2 - 24n + 9\). The original problem thus becomes a simpler quadratic equation, which makes further algebraic operations more straightforward.

Quadratic Equations

Quadratic equations are polynomial equations of degree two with the general form \(ax^2 + bx + c = 0\). They represent parabolas when graphed on a coordinate axis.

In our exercise, after the polynomial expansion of \((4n - 3)^2\), we derived \(16n^2 - 24n + 9\), which is a quadratic equation. This expression is characterized by:

• A leading term: \(16n^2\)
• A linear term: \(-24n\)
• A constant term: \(9\)

These components together make the quadratic equation. Solving such an equation might involve finding its roots, using methods like factoring, completing the square, or applying the quadratic formula. Each method has its advantages depending on the form of the equation and makes it versatile in solving problems across various contexts.

Understanding the breakdown into a quadratic form helps in proceeding with solutions, predicting the behavior of the equation graphically, and applying further operations efficiently.

Algebraic Expressions

Algebraic expressions consist of variables, coefficients, and constants arranged in a meaningful formula. They form the basis of algebra and are used to represent relationships between numbers. In the exercise, the original expression \((4n - 3)^2\) involves several algebraic entities:

• Variables: like \(n\), which can represent numbers
• Coefficients: the number \(4\) associated with \(n\)
• Constants: like \(-3\) in the expression

This algebraic expression can subsequently be expanded and simplified using algebraic rules like the binomial theorem. Expanding and simplifying helps in managing algebraic expressions, making them easier to handle, especially when solving real-world problems.

The importance of mastering algebraic expressions lies in their applicability. They allow us to model and understand numerical relationships, solve equations, and analyze mathematical and empirical data effectively. By transforming \((4n - 3)^2\) into \(16n^2 - 24n + 9\), the expression is in a form that is simpler and more usable in computational and analytical tasks.