Understanding And Factoring Quadratic Expressions

Introduction

In mathematics, a quadratic expression is a polynomial of degree two. It generally takes the form of ax² + bx + c, where a, b, and c are constants, and x is the variable. When given a quadratic expression like x² + 3x + 4, the goal is to factor it into two binomials. This article will focus on how to factor expressions like these, specifically looking at the example of x² + 3x + 4.

Step 1: Find the factors of the leading coefficient and constant term

The first step in factoring a quadratic expression is to determine the factors of the leading coefficient (the coefficient of the x² term) and the constant term (the term without an x). For the expression x² + 3x + 4, the leading coefficient is 1 and the constant term is 4. The factors of 1 are simply 1 and 1, while the factors of 4 are 1 and 4 or 2 and 2.

Step 2: Determine which factor pairs add up to the coefficient of the x-term

Next, we need to find the factor pairs of the constant term that add up to the coefficient of the x-term. For x² + 3x + 4, the coefficient of the x-term is 3. We need to find two numbers that multiply to 4 and add up to 3. The factor pairs of 4 are (1, 4) and (2, 2). Of these, only (1, 4) adds up to 3. Therefore, we can write the expression as (x + 1)(x + 4).

Step 3: Check your answer

After factoring the expression, it is important to check your answer. You can do this by multiplying out the two binomials you found in step 2 and making sure it equals the original expression. In this case, (x + 1)(x + 4) = x² + 5x + 4, which is not the same as the original expression. So, we need to go back and try again.

Step 4: Adjust your factor pairs

Since our first attempt did not work, we need to adjust the factor pairs we found in step 2. We know that (1, 4) does not work, so we can try (2, 2). This gives us the binomial (x + 2) twice, or (x + 2)².

Step 5: Check your answer again

To check this answer, we can multiply out (x + 2)² using the FOIL method. This gives us x² + 4x + 4. This is the same as the original expression, so we know we have factored it correctly.

Conclusion

In conclusion, factoring quadratic expressions can seem intimidating at first, but by following the steps outlined above, it can become much more manageable. When factoring x² + 3x + 4, we found the factor pairs of the constant term and then determined which pairs added up to the coefficient of the x-term. After adjusting our factor pairs, we were able to find the correct binomial factorization, which was (x + 2)². Remember to always check your answer by multiplying out your binomials and making sure they equal the original expression.