Understanding Factorization and Quadratic Equations
Factorization is a key concept in algebra, particularly when dealing with quadratic equations. A quadratic equation is a polynomial of degree two, and can be expressed in the form ax^2 bx c 0. The process of factorization helps us rewrite the equation in a simpler form that can be easily solved. In this article, we’ll explore how to factor the expression 3G^2 - 48, breaking it down step by step.Identifying the Expression to Factor
The expression we are dealing with is 3G^2 - 48. This is a quadratic expression that can be factored using the difference of squares formula. Let’s first simplify and identify the terms within the expression.Step 1: Simplification
Before we factor the expression, let's simplify it a bit to understand its components better. Notice that 48 is divisible by 3, so we can factor out the greatest common factor (GCF). 3G^2 - 48 3G^2 - 3 * 16 This simplifies the expression to: 3G^2 - 3 * 16 or 3G^2 - 16Step 2: Factoring by the Difference of Squares
The expression 3G^2 - 16 is a difference of squares. The difference of squares formula states that a^2 - b^2 (a - b)(a b). We need to rewrite 3G^2 - 16 in this form. In this case, we have 3G^2 and 16. Let's identify a and b: G^2 a^2 16 b^2 Here, (a G) and (b 4) because (4^2 16). Using the difference of squares formula, we get: 3G^2 - 16 3(G^2 - 4^2) 3(G - 4)(G 4) However, the task specifies a different factorization method involving a common factor. Let’s break the original expression further into simpler steps.Step 3: Factoring with a Common Factor
First, we factor out the greatest common factor (GCF) from the expression. In this case, the GCF of the two terms (3G^2 and 48) is 3. 3G^2 - 48 3(G^2 - 16) Now, we recognize that 16 is a perfect square (4^2), so we can rewrite the expression inside the parentheses using the difference of squares formula. G^2 - 16 G^2 - 4^2 (G - 4)(G 4) Therefore, the complete factorization of 3G^2 - 48 is: 3(G - 4)(G 4)Conclusion
In summary, the factorization of the expression 3G^2 - 48 can be achieved by first identifying the greatest common factor and then applying the difference of squares formula. The final factored form is 3(G - 4)(G 4).FAQs
What is the difference of squares formula?The difference of squares formula is a^2 - b^2 (a - b)(a b). This formula helps in factoring expressions that are the difference of two perfect squares.
How do you factor out the greatest common factor?To factor out the greatest common factor, you first identify the common factor in the terms of the expression. For example, in 3G^2 - 48, the GCF is 3.
Can every quadratic expression be factored?Not all quadratic expressions can be factored using integers. However, the methods discussed here can be applied to many common cases. If factoring with integers is difficult, you might need to use the quadratic formula or completing the square.