Factorize X^2 - X - 6: A Step-By-Step Guide

Here's a comprehensive guide on how to factorize the quadratic expression ( X^2 - X - 6 ) with step-by-step instructions:

Understanding Quadratic Expressions 🧩

<div style="text-align: center;"> <img src="https://tse1.mm.bing.net/th?q=quadratic expressions" alt="Quadratic expressions"> </div>

Before diving into factorization, let's quickly understand what we're dealing with:

• Quadratic expressions are polynomials of degree 2, expressed in the form ( ax^2 + bx + c ).
• In our case, ( X^2 - X - 6 ), where ( a = 1 ), ( b = -1 ), and ( c = -6 ).

Why Factorize? 🔍

• Simplification: Factorization makes expressions easier to work with, especially when solving equations or simplifying complex terms.
• Solving Equations: It's crucial for solving quadratic equations by setting each factor to zero.
• Simplify Expressions: It can reveal common factors or patterns that help in further simplification or solving.

Step 1: Identify the Product of the Leading Coefficient and the Constant 🔢

<div style="text-align: center;"> <img src="https://tse1.mm.bing.net/th?q=polynomial factorization" alt="Polynomial factorization"> </div>

The leading coefficient is 1, and the constant term is -6:

• Product: (1 \times -6 = -6)

<p class="pro-note">📝 Note: In this case, (a = 1), simplifying the factorization process.</p>

Step 2: Find Factors of the Constant 🔍

List all pairs of factors of -6:

• (1, -6)
• (-1, 6)
• (2, -3)
• (-2, 3)
• (3, -2)
• (-3, 2)

<p class="pro-note">📝 Note: Both positive and negative pairs are considered as the sign of the coefficient (b) is negative.</p>

Step 3: Choose the Pair Whose Sum Equals the Linear Coefficient 🧮

The linear coefficient (b) is -1:

• From the pairs, (3 + (-2) = 1), which doesn't match our requirement.
• The correct pair is (-3 + 2 = -1), which is exactly what we need.

Step 4: Rewrite the Middle Term Using the Chosen Factors 🏃‍♂️

Replace (-X) with (2X - 3X):

• (X^2 - X - 6 \rightarrow X^2 + 2X - 3X - 6)

<p class="pro-note">📝 Note: This step doesn't change the value of the polynomial; it's just a way to group and factor out common terms.</p>

Step 5: Group and Factor Out Common Terms 👥

• ( (X^2 + 2X) - (3X + 6) )

Now, factor out the common terms from each group:

• From the first group: ( X(X + 2) )
• From the second group: ( -3(X + 2) )

Step 6: Factor by Grouping 🔍

Now that we have:

• ( X(X + 2) - 3(X + 2) )

Factor out the common binomial ((X + 2)):

• ( (X + 2)(X - 3) )

Conclusion

We've successfully factorized ( X^2 - X - 6 ) into:

• ( (X + 2)(X - 3) )

This factorization tells us that the solutions to the equation ( X^2 - X - 6 = 0 ) are ( X = -2 ) or ( X = 3 ), and it simplifies the expression for further algebraic manipulation.

Here are some frequently asked questions regarding this topic:

<div class="faq-section"> <div class="faq-container"> <div class="faq-item"> <div class="faq-question"> <h3>Why do we need to factorize quadratic expressions?</h3> <span class="faq-toggle">+</span> </div> <div class="faq-answer"> <p>Factorization helps in solving quadratic equations, simplifying expressions, and understanding the structure of polynomials.</p> </div> </div> <div class="faq-item"> <div class="faq-question"> <h3>What if the leading coefficient is not 1?</h3> <span class="faq-toggle">+</span> </div> <div class="faq-answer"> <p>If (a \neq 1), the process becomes more complex. You would need to factor out (a) first or use methods like AC method or trial and error.</p> </div> </div> <div class="faq-item"> <div class="faq-question"> <h3>Can quadratic expressions always be factorized?</h3> <span class="faq-toggle">+</span> </div> <div class="faq-answer"> <p>Not all quadratic expressions can be factorized over the integers, but they can always be solved using the quadratic formula.</p> </div> </div> </div> </div>