Factoring Quadratic Equations: Solve X² + 4x - 21
Hey math enthusiasts! Today, we're diving into the world of factoring quadratic equations, specifically tackling the equation x² + 4x - 21 = (x - _)(x + _). Factoring might seem a bit tricky at first, but trust me, with a few simple steps, you'll be breaking down these equations like a pro. This guide will walk you through the process, explain the key concepts, and give you the tools you need to conquer this type of problem. So, grab your pencils, and let's get started. We will explore step-by-step how to solve the quadratic equation and give a comprehensive guide on how to factor it. This guide ensures that everyone can understand and solve the problem.
Understanding Quadratic Equations and Factoring
Before we jump into the equation, let's make sure we're all on the same page about what quadratic equations and factoring actually are. A quadratic equation is an equation of the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. These equations always have an x² term, which gives them their characteristic curved shape when graphed. Factoring, on the other hand, is the process of breaking down a quadratic expression into a product of simpler expressions (usually two binomials). Think of it like taking a number and finding its prime factors. For example, the factors of 12 are 2, 2, and 3, because 2 x 2 x 3 = 12. In the case of quadratic equations, we want to find two binomials that, when multiplied together, give us the original quadratic expression. The goal here is to rewrite the quadratic expression as a product of two linear expressions in the form of (x + p)(x + q). Understanding the concepts of quadratics and how factoring works is crucial before we start solving. Mastering the process takes practice, but the rewards are well worth the effort. Let's make sure we have a solid grasp of these foundations before we proceed. The key is to find those two special numbers that fit perfectly into the binomials.
When we factor, we're essentially reversing the process of expanding binomials (using the FOIL method, which stands for First, Outer, Inner, Last). So, when we see x² + 4x - 21, we need to find two numbers that multiply to give us -21 (the constant term) and add up to give us 4 (the coefficient of the x term). If you're a beginner, don't worry, it might seem difficult at first, but with practice, it becomes second nature. Let's break down the process in detail, making sure everyone can follow along. Remember to keep in mind the signs of the numbers. That's a common area where people get tripped up. Factoring can be a puzzle, but a fun one! So, let's put on our detective hats and start solving.
Step-by-Step Solution of
Alright, let's get down to business and solve x² + 4x - 21 = (x - _)(x + _). The key to solving this equation is to find two numbers that meet two conditions: they multiply to -21 and add up to 4. Let's start by listing the factor pairs of -21. Remember, since the product is negative, one number must be positive and the other must be negative.
- 1 and -21
- -1 and 21
- 3 and -7
- -3 and 7
Now, let's check which of these pairs adds up to 4.
- 1 + (-21) = -20
- -1 + 21 = 20
- 3 + (-7) = -4
- -3 + 7 = 4
There it is! The pair -3 and 7 adds up to 4. That means these are the numbers we need. Now, we can plug these numbers into our factored form. Therefore, x² + 4x - 21 = (x - 3)(x + 7). That's it, guys! We've successfully factored the quadratic equation. Wasn't that fun? The process might feel a little cumbersome at first, but with practice, you will become faster at finding the factors. Understanding the relationship between the coefficients and the factors is key. Now that we have the factored form, we can easily find the solutions (the values of x that make the equation equal to zero). In this case, x = 3 and x = -7. This means that if we substitute 3 or -7 for x in the original equation, the equation will be equal to zero. Give it a try if you want to check your answer.
Verification and Further Practice
To make sure we've done everything correctly, it's always a good idea to check our work. We can do this by expanding the factored form and seeing if it matches the original quadratic equation. Let's expand (x - 3)(x + 7) using the FOIL method:
- First: x * x = x²
- Outer: x * 7 = 7x
- Inner: -3 * x = -3x
- Last: -3 * 7 = -21
Combining these terms, we get x² + 7x - 3x - 21 = x² + 4x - 21. And what do you know? This matches the original equation perfectly. That's how you know you have the right solution. Factoring can seem difficult at first, but it gets easier with each problem you solve. This verification step provides you with added confidence in your solution, showing you that your answer aligns with the initial quadratic equation. When you work with quadratics, there are often two solutions. Also, the ability to solve the equation is the ability to find the roots, or x-intercepts, of the related quadratic function when graphed. Practice makes perfect, and the more you practice, the better you will become at recognizing patterns and finding the correct factors quickly. Take the time to practice with different examples and soon, you will become a factoring expert.
To solidify your understanding, let's practice another example. Suppose you need to factor x² + 8x + 15. The factor pairs for 15 are (1, 15), (3, 5), (-1, -15), (-3, -5). The pair (3, 5) adds up to 8. Therefore, x² + 8x + 15 = (x + 3)(x + 5). Practice is the key to mastering this skill. If the sign of the constant term is negative, then one of the factors must be negative. If the sign of the constant term is positive and the sign of the linear term is positive, then both factors must be positive. Remember that with enough practice, you'll be able to solve these problems quickly and confidently.
Advanced Factoring Techniques
While this guide has focused on simple quadratic equations, it's important to know there are more advanced techniques when dealing with more complex problems. Sometimes, the coefficient of the x² term isn't 1. For example, equations like 2x² + 5x + 2 require a slightly different approach. In these cases, you might use techniques like the