What Is The Factored Form Of X2 + X + 2?

Introduction

When studying algebra, students will come across polynomial equations that require them to factorize them. One such equation is x2 + x + 2. In this article, we will explain what factoring is and provide a step-by-step solution to finding the factored form of x2 + x + 2.

What is Factoring?

Factoring is a process of breaking down a polynomial equation into its simplest form. The simplest form is when the equation is expressed as a product of its factors. For instance, x2 + x can be factored into x(x+1).

How to Factorize x2 + x + 2

To factorize x2 + x + 2, we need to find two numbers that multiply to 2 and add up to 1. These two numbers will be the factors of the equation.

Step 1: Find the Factors of 2

The first step is to find the factors of 2, which are 1 and 2.

Step 2: Determine the Right Combination

We need to determine which combination of these factors will add up to 1. We can do this by trial and error. 1 + 2 = 3, which is not equal to 1. 2 + 1 = 3, which is not equal to 1. Thus, there is no combination of factors that add up to 1. Therefore, x2 + x + 2 cannot be factored using real number factors alone.

Step 3: Using Complex Numbers

However, it is still possible to factorize x2 + x + 2 using complex numbers. We can express it in the form of (x + a)(x + b), where a and b are complex numbers.

Step 4: Finding the Complex Numbers

To find the complex numbers a and b, we can use the quadratic formula. x = (-b ± sqrt(b2 - 4ac))/2a In this case, a = 1, b = 1, and c = 2. x = (-1 ± sqrt(1 - 4(1)(2)))/2(1) x = (-1 ± sqrt(-7))/2 Therefore, the complex numbers a and b are: a = (-1 + sqrt(-7))/2 b = (-1 - sqrt(-7))/2

Step 5: Expressing in Factored Form

Now that we have the complex numbers a and b, we can express x2 + x + 2 in factored form as: (x + (-1 + sqrt(-7))/2)(x + (-1 - sqrt(-7))/2)

Conclusion

In conclusion, x2 + x + 2 cannot be factored using real number factors alone. However, it can be factorized using complex numbers. Factoring is an essential skill in algebra and is used in many applications, such as solving quadratic equations and simplifying expressions.