Problem 50 - Algebra and Trigonometry

Question

Multiply the algebraic expressions using a Special Product Formula, and simplify. \((x-2)^{2}\)

Step-by-Step Solution

Answer

The expression \((x-2)^2\) simplifies to \(x^2 - 4x + 4\).

Step 1: Recognize the Special Product Formula

The given expression is \((x-2)^2\). This is a square of a binomial, which means we can use the special product formula: \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = x\) and \(b = 2\).Our goal is to apply this formula to the expression.

Step 2: Apply the Formula

Substituting \(a = x\) and \(b = 2\) into \((a - b)^2 = a^2 - 2ab + b^2\), we calculate:\(a^2 = x^2\)\(-2ab = -2 imes x imes 2 = -4x\)\(b^2 = 2^2 = 4\)Putting it together, \((x - 2)^2 = x^2 - 4x + 4\).

Step 3: Simplify the Expression

The expression \(x^2 - 4x + 4\) is already simplified as there are no like terms to combine. This is the simplified result of the original expression \((x-2)^2\) using the special product formula.

Key Concepts

Binomial Squaring

Squaring a binomial might sound tricky, but it's a powerful and often used technique in algebra. The term typically involves something like a binomial of the form \((a - b)\) or \((a + b)\). When we square these binomials, we expand them as follows:

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

These are called special product formulas and are really just shortcuts for multiplying the binomial by itself. They are essential to remember because they let us expand expressions efficiently.
Take the expression given in the exercise \((x-2)^2\). Using the formula directly saves time, allowing you to write:\[a^2 - 2ab + b^2\] as \[x^2 - 4x + 4.\] It takes less than a minute to solve when you know the formula! Understanding these expansions not only helps in simplifying expressions but also boosts your confidence in tackling more complex algebra problems.

Algebraic Expressions

Algebraic expressions are everywhere in math, and it's important to understand how to work with them. These expressions are formed by combining numbers, variables, and operation signs like addition, subtraction, multiplication, and division.
In the expression from our exercise, \((x-2)^2\), both \(x\) and \(-2\) are terms of the algebraic expression, and those are linked by the subtraction operator. When we square this binomial, it teaches us two key ideas:

• Understanding the structure of expressions
• Being able to manage variables and constants together

This style of writing breaks down complicated formulae into understandable pieces.
Learning to identify patterns in algebraic expressions leads to quicker problem-solving and builds a stronger foundation for future mathematical challenges.

Polynomial Simplification

Simplification of polynomials is a process used to make expressions more manageable. The simpler the expression, the easier it is to work with, which is why it often follows the application of special product formulas like the ones in binomial squaring.
Let's see how it works in our example.
After applying the special product formula \((x-2)^2\), the expanded expression is \(x^2 - 4x + 4\). Unlike some expressions, this one is already simplified after expansion because there are no terms left to combine.

• No like terms need to be grouped together.
• Each part of the expression is distinct.

Understanding simplification ensures you have the cleanest form of an expression and is a valuable skill, especially in more advanced mathematics, where expressions could become even more complex.