Factoring Quadratic Expressions A Step-by-Step Guide To Solving X^2 + X - 72

Understanding the Basics of Factoring Quadratics

Before we dive into the specifics of factoring x^2 + x - 72, it's crucial to understand the basic principles behind factoring quadratic expressions. Factoring is essentially the reverse process of expanding or multiplying out expressions. When we expand (x + a)(x + b), we use the distributive property (often remembered by the acronym FOIL – First, Outer, Inner, Last) to get x^2 + bx + ax + ab, which simplifies to x^2 + (a + b)x + ab. Therefore, when we factor a quadratic expression, we are looking for two numbers, 'a' and 'b', that satisfy two conditions: their product should equal the constant term (c), and their sum should equal the coefficient of the x term (b). In our case, for the expression x^2 + x - 72, we need to find two numbers that multiply to -72 and add up to 1. This might seem like a daunting task initially, but with a systematic approach and practice, it becomes much more manageable. Understanding this fundamental relationship between the coefficients and the constants is the key to successful factoring. By breaking down the problem into smaller, more manageable steps, we can effectively factor quadratic expressions of varying complexity. This foundational knowledge will empower you to tackle a wide range of algebraic problems with confidence.

Identifying the Factors of -72

In this crucial step, we focus on identifying the factors of -72, which is the constant term in our quadratic expression x^2 + x - 72. The factors of a number are the integers that divide evenly into that number. Since we are looking for factors of a negative number, we know that one factor must be positive, and the other must be negative. To find the correct pair of factors, we can systematically list the factor pairs of 72 and consider their signs. The factor pairs of 72 are (1, 72), (2, 36), (3, 24), (4, 18), (6, 12), and (8, 9). Now, we need to consider the combinations of these pairs where one number is positive, and the other is negative. This gives us the following possibilities: (-1, 72), (1, -72), (-2, 36), (2, -36), (-3, 24), (3, -24), (-4, 18), (4, -18), (-6, 12), (6, -12), (-8, 9), and (8, -9). Each of these pairs multiplies to -72, but we need to find the specific pair that also adds up to the coefficient of our x term, which is 1. This systematic approach ensures that we consider all possible factor pairs, increasing our chances of finding the correct combination. Without a systematic method, it can be easy to overlook a crucial factor pair, leading to incorrect factoring. Therefore, meticulously listing out the factors is a vital step in the process.

Finding the Correct Pair: Summing to 1

After identifying the factor pairs of -72, the next critical step is to find the pair that sums to 1, which is the coefficient of the 'x' term in our quadratic expression x^2 + x - 72. We have already listed the factor pairs as (-1, 72), (1, -72), (-2, 36), (2, -36), (-3, 24), (3, -24), (-4, 18), (4, -18), (-6, 12), (6, -12), (-8, 9), and (8, -9). Now, we need to check the sum of each pair. Let's go through them one by one:

• -1 + 72 = 71
• 1 + (-72) = -71
• -2 + 36 = 34
• 2 + (-36) = -34
• -3 + 24 = 21
• 3 + (-24) = -21
• -4 + 18 = 14
• 4 + (-18) = -14
• -6 + 12 = 6
• 6 + (-12) = -6
• -8 + 9 = 1
• 8 + (-9) = -1

From this list, we can see that the pair -8 and 9 sums to 1. This is the pair we need because it satisfies both conditions: it multiplies to -72 and adds up to 1. This process of systematically checking the sums highlights the importance of precision in factoring. A small error in addition or subtraction can lead to choosing the wrong pair, resulting in an incorrect factored expression. Therefore, taking the time to carefully verify each sum is essential for accurate factoring. With the correct pair identified, we are now ready to rewrite the quadratic expression in its factored form.

Rewriting the Expression in Factored Form

Having identified the correct pair of numbers, -8 and 9, we can now rewrite the quadratic expression x^2 + x - 72 in its factored form. The factored form will be (x + a)(x + b), where 'a' and 'b' are the numbers we found. In this case, a = -8 and b = 9. Therefore, the factored form of the expression is (x - 8)(x + 9). To verify that this is indeed the correct factorization, we can expand the factored form using the distributive property (FOIL method): (x - 8)(x + 9) = x(x) + x(9) - 8(x) - 8(9) = x^2 + 9x - 8x - 72 = x^2 + x - 72. This expansion confirms that our factored form is equivalent to the original expression. Rewriting the expression in factored form is the culmination of the factoring process, providing a more simplified and insightful representation of the quadratic expression. The factored form is particularly useful for solving quadratic equations, as it allows us to easily identify the roots or solutions of the equation. Furthermore, the factored form can reveal important information about the graph of the quadratic function, such as its x-intercepts. Thus, mastering the skill of rewriting expressions in factored form is a valuable asset in algebra and beyond.

Final Answer: (x - 8)(x + 9)

In conclusion, we have successfully factored the quadratic expression x^2 + x - 72 and rewritten it in the factored form (x - 8)(x + 9). This process involved several key steps, including understanding the basics of factoring quadratics, identifying the factors of the constant term (-72), finding the pair of factors that sums to the coefficient of the 'x' term (1), and finally, rewriting the expression using these factors. We meticulously identified all factor pairs of -72 and systematically checked their sums to find the correct pair, -8 and 9. This systematic approach ensured accuracy and helped us avoid common factoring errors. We then verified our solution by expanding the factored form (x - 8)(x + 9), which confirmed that it is equivalent to the original expression x^2 + x - 72. The ability to factor quadratic expressions is a crucial skill in algebra, with applications in solving equations, simplifying expressions, and understanding the behavior of quadratic functions. By mastering this skill, you gain a deeper understanding of algebraic concepts and enhance your problem-solving abilities. The factored form (x - 8)(x + 9) provides valuable insights into the roots of the corresponding quadratic equation and the x-intercepts of the quadratic function's graph. Therefore, understanding and practicing factoring techniques is essential for success in algebra and higher-level mathematics.

Therefore, the values are a = -8 and b = 9.