Factoring is a fundamental concept in algebra that involves breaking down an algebraic expression into simpler components. Finding the factored form of a quadratic expression, such as x^2 + x – 2, can be a challenging task for some students. In this article, we will explore the step-by-step process of factoring the quadratic expression x^2 + x – 2 and understanding its factored form.
Understanding the Quadratic Expression
Before we delve into factoring the expression x^2 + x – 2, let’s break down the components of the quadratic expression:
- The expression x^2 represents a squared term, where x is a variable raised to the power of 2.
- The term x is a linear term, which is a first-degree polynomial.
- The constant term -2 is a numerical value without any variable attached to it.
Factors of the Quadratic Expression
To find the factored form of the quadratic expression x^2 + x – 2, we need to consider the factors of the quadratic expression that will help us simplify it. In this case, we are looking for two factors of -2 that add up to 1 (the coefficient of the linear term).
Step-by-Step Process
Now, let’s walk through the step-by-step process of factoring the quadratic expression x^2 + x – 2:
- Identify the coefficients: The coefficients of the quadratic expression are a = 1, b = 1, and c = -2 in the general form ax^2 + bx + c.
- Find the factors of c: We need to find two numbers that multiply to -2 (ac) and add up to 1 (b).
- Write down the factors: The factors of -2 that meet the criteria are -2 and 1.
- Split the middle term: Rewrite the linear term (x) using the factors obtained: x^2 – 2x + 3x – 2.
- Factor by grouping: Group the terms into two pairs and factor out the common factors: x(x – 2) + 1(x – 2).
- Write the factored form: Combine the factored terms to obtain the final factored form: (x + 1)(x – 2).
Verification and Expansion
It is important to verify our factored form by expanding it to ensure that we obtain the original quadratic expression. Let’s expand (x + 1)(x – 2) and simplify the expression:
(x + 1)(x – 2) = x(x) + x(-2) + 1(x) + 1(-2)
= x^2 – 2x + x – 2
= x^2 – x – 2
By expanding the factored form, we have successfully obtained the original quadratic expression x^2 – x – 2, confirming that our factored form is correct.
Conclusion
Factoring a quadratic expression like x^2 + x – 2 requires a systematic approach to identify the factors that will simplify the expression. By following the step-by-step process and understanding the factors involved, you can efficiently find the factored form of the quadratic expression. Practice and familiarity with factoring techniques will enhance your ability to factor more complex quadratic expressions in the future.
