How To Factorize Bracket Power 2

How to Factorize Expressions with Brackets Raised to the Power of 2

Factoring expressions, particularly those involving brackets raised to the power of 2, is a crucial algebraic skill. Mastering this technique is essential for simplifying expressions, solving equations, and tackling more advanced mathematical concepts. This comprehensive guide will walk you through the process step-by-step, covering various scenarios and providing clear examples.

Understanding the Concept

Before diving into the techniques, let's establish a firm grasp of the fundamental concept. When we have an expression like (a + b)² or (a - b)², we're essentially dealing with a perfect square trinomial. This means the expression, when expanded, will result in a trinomial (three terms) where two of the terms are perfect squares, and the remaining term is twice the product of the square roots of the perfect square terms.

The Formulae: Your Key to Success

Remember these key formulae, as they form the foundation for factorizing expressions with brackets raised to the power of 2:

• (a + b)² = a² + 2ab + b²
• (a - b)² = a² - 2ab + b²

These formulae allow you to expand the squared brackets. However, we're interested in factorizing, which is the reverse process – moving from the expanded trinomial back to the squared bracket.

Step-by-Step Factorization Guide

Let's illustrate the factorization process with a step-by-step example:

Example: Factorize x² + 6x + 9

Step 1: Identify Perfect Squares

Check if the first and last terms are perfect squares. In this case:

• x² is the square of x (x²)
• 9 is the square of 3 (3²)

Step 2: Check the Middle Term

The middle term (6x) should be twice the product of the square roots of the first and last terms. Let's verify:

2 * x * 3 = 6x

This matches the middle term.

Step 3: Apply the Formula

Since the conditions are met, we can apply the formula (a + b)² = a² + 2ab + b². Here, a = x and b = 3:

x² + 6x + 9 = (x + 3)²

Therefore, the factorized form of x² + 6x + 9 is (x + 3)².

Handling More Complex Scenarios

The process remains similar even when dealing with more complex expressions. Consider this example:

Example: Factorize 4x² - 12x + 9

Step 1: Identify Perfect Squares

• 4x² is the square of 2x ((2x)²)
• 9 is the square of 3 (3²)

Step 2: Check the Middle Term

The middle term (-12x) should be twice the product of the square roots of the first and last terms:

2 * 2x * 3 = 12x

Note the negative sign!

Step 3: Apply the Formula

We'll use the formula (a - b)² = a² - 2ab + b², where a = 2x and b = 3:

4x² - 12x + 9 = (2x - 3)²

Practice Makes Perfect

The key to mastering factorization is practice. Work through various examples, gradually increasing the complexity of the expressions. Look for perfect squares and meticulously check the middle term. With consistent practice, you’ll quickly become proficient in factorizing expressions with brackets raised to the power of 2. Remember to always double-check your work by expanding your factored expression to confirm it matches the original expression.