Factoring X^2 + 18x + 77 A Step-by-Step Guide

Factoring quadratic expressions is a fundamental skill in algebra, with applications ranging from solving equations to simplifying complex expressions. In this article, we will delve into the process of factoring the quadratic expression x^2 + 18x + 77. We will explore the underlying principles, step-by-step methods, and common strategies for tackling such problems. By the end of this guide, you will have a solid understanding of how to factor quadratic expressions effectively.

Understanding Quadratic Expressions

Before we dive into the specific example, let's first establish a clear understanding of what quadratic expressions are. A quadratic expression is a polynomial of degree two, meaning the highest power of the variable is two. The general form of a quadratic expression is ax^2 + bx + c, where a, b, and c are constants, and x is the variable. The coefficient a determines the shape of the parabola when the quadratic expression is graphed, b influences the position of the axis of symmetry, and c represents the y-intercept.

In our case, the quadratic expression is x^2 + 18x + 77. Comparing this to the general form, we can identify the coefficients: a = 1, b = 18, and c = 77. These coefficients play a crucial role in the factoring process. Understanding the relationship between these coefficients and the factored form is key to mastering this skill. The goal of factoring is to rewrite the quadratic expression as a product of two linear expressions, typically in the form (x + p)(x + q), where p and q are constants. When we expand this product, we get x^2 + (p + q)x + pq. This expanded form gives us a direct comparison to the original quadratic expression, allowing us to determine the values of p and q.

The Factoring Process

1. Identify the Coefficients

The first step in factoring the quadratic expression x^2 + 18x + 77 is to identify the coefficients a, b, and c. As we mentioned earlier, in this case, a = 1, b = 18, and c = 77. These values are essential for determining the factors of the quadratic expression. The coefficient a (which is 1 in this case) simplifies the factoring process somewhat, but the principles remain the same for other values of a. The key is to find two numbers that satisfy specific conditions related to b and c.

2. Find Two Numbers

Next, we need to find two numbers, let's call them p and q, such that their product equals the constant term c (which is 77) and their sum equals the coefficient b (which is 18). In other words, we are looking for p and q such that p * q = 77 and p + q = 18. This is the most critical step in the factoring process, and it often involves some trial and error. We need to consider the factors of 77 and see which pair adds up to 18. The factors of 77 are 1 and 77, and 7 and 11. Clearly, 7 and 11 satisfy the conditions since 7 * 11 = 77 and 7 + 11 = 18.

3. Write the Factored Form

Once we have found the numbers p and q, we can write the factored form of the quadratic expression. In our case, p = 7 and q = 11. Therefore, the factored form is (x + 7)(x + 11). This means that the quadratic expression x^2 + 18x + 77 can be rewritten as the product of the two linear expressions (x + 7) and (x + 11). This factored form is extremely useful for solving quadratic equations, as it allows us to find the values of x that make the expression equal to zero.

4. Verify the Solution

To ensure that we have factored the quadratic expression correctly, we should always verify our solution by expanding the factored form and comparing it to the original expression. Expanding (x + 7)(x + 11) using the distributive property (also known as the FOIL method) gives us:

(x + 7)(x + 11) = x * x + x * 11 + 7 * x + 7 * 11 = x^2 + 11x + 7x + 77 = x^2 + 18x + 77

Since the expanded form matches the original quadratic expression x^2 + 18x + 77, we can confidently say that our factoring is correct. This step is crucial for avoiding errors and ensuring the accuracy of our work.

Alternative Method: The Quadratic Formula

While factoring is a powerful technique, it's not always straightforward, especially when the coefficients are large or the quadratic expression is not easily factorable. In such cases, the quadratic formula provides a reliable alternative for finding the roots (or zeros) of the quadratic expression. The quadratic formula is given by:

x = (-b ± √(b^2 - 4ac)) / (2a)

where a, b, and c are the coefficients of the quadratic expression ax^2 + bx + c = 0. For our expression x^2 + 18x + 77, we have a = 1, b = 18, and c = 77. Plugging these values into the quadratic formula, we get:

x = (-18 ± √(18^2 - 4 * 1 * 77)) / (2 * 1) x = (-18 ± √(324 - 308)) / 2 x = (-18 ± √16) / 2 x = (-18 ± 4) / 2

This gives us two solutions:

x = (-18 + 4) / 2 = -14 / 2 = -7 x = (-18 - 4) / 2 = -22 / 2 = -11

These roots correspond to the factors (x + 7) and (x + 11), which confirms our earlier factoring result.

When is a Polynomial Prime?

In some cases, a quadratic expression cannot be factored into simpler linear expressions with integer coefficients. Such expressions are called prime polynomials. A polynomial is considered prime if it cannot be factored over the set of integers. This means that there are no two binomials with integer coefficients that, when multiplied, will result in the given polynomial. Determining whether a polynomial is prime is an important aspect of factoring.

To determine if a quadratic expression is prime, we can check the discriminant, which is the part of the quadratic formula under the square root sign: b^2 - 4ac. If the discriminant is negative or not a perfect square, then the quadratic expression is prime (over the integers). In our example, the discriminant is 18^2 - 4 * 1 * 77 = 324 - 308 = 16, which is a perfect square. This indicates that the quadratic expression is factorable, as we have already shown.

However, if the discriminant were negative, it would mean that the roots are complex numbers, and the quadratic expression cannot be factored using real numbers. If the discriminant were positive but not a perfect square, the roots would be irrational, and the quadratic expression would still be considered prime over the integers.

Common Mistakes and How to Avoid Them

Factoring quadratic expressions can be challenging, and it's common to make mistakes, especially when starting out. One common mistake is incorrectly identifying the factors of the constant term or the signs of the factors. For example, when factoring x^2 + 18x + 77, a mistake might be made in finding the correct pair of numbers that multiply to 77 and add up to 18. It's crucial to double-check these numbers to ensure they satisfy both conditions.

Another common mistake is forgetting to verify the solution by expanding the factored form. This step is essential for catching errors and ensuring accuracy. Always take the time to multiply the factored binomials back together to see if they match the original quadratic expression.

Additionally, some students may struggle with the concept of prime polynomials. It's important to understand that not all quadratic expressions can be factored, and sometimes the correct answer is that the polynomial is prime. Using the discriminant can help determine if a quadratic expression is prime or not.

Conclusion

Factoring the quadratic expression x^2 + 18x + 77 involves finding two numbers that multiply to 77 and add up to 18. By following a step-by-step approach, we identified these numbers as 7 and 11, leading to the factored form (x + 7)(x + 11). We also discussed the quadratic formula as an alternative method for finding the roots and explored the concept of prime polynomials. Mastering factoring is a crucial skill in algebra, and with practice, you can become proficient at factoring various quadratic expressions.

Remember to always verify your solutions and be mindful of common mistakes. By understanding the underlying principles and applying them diligently, you can confidently tackle factoring problems and enhance your algebraic skills.