Solving Quadratic Equations By Factoring 16x² + 2x - 3 = 0
Hey guys! Let's dive into solving a quadratic equation by factoring. Factoring is a super useful skill in algebra, and it’s like reverse engineering multiplication. We’re going to take a quadratic expression and break it down into two binomial expressions that, when multiplied together, give us the original quadratic. It might sound tricky, but once you get the hang of it, it's pretty straightforward. We'll be tackling the equation 16x² + 2x - 3 = 0. This type of equation is a classic example where factoring can help us find the solutions. The solutions are the values of x that make the equation true, meaning when you plug them back into the equation, both sides are equal.
Understanding Quadratic Equations
Before we jump into the nitty-gritty, let's quickly recap what a quadratic equation is. A quadratic equation is generally written in the form ax² + bx + c = 0, where a, b, and c are constants, and x is the variable we're trying to solve for. The 'a' term is the coefficient of the x² term, the 'b' term is the coefficient of the x term, and 'c' is the constant term. In our equation, 16x² + 2x - 3 = 0, a is 16, b is 2, and c is -3. Recognizing these coefficients is the first step in figuring out how to factor the equation. Remember, the goal of solving any equation is to find the value(s) of the variable that make the equation a true statement. For quadratic equations, this often means finding two solutions because of the squared term. These solutions are also known as roots or zeros of the equation. Factoring is just one method we can use to find these solutions. Other methods include using the quadratic formula or completing the square, but factoring is often the quickest and most elegant method when it works. So, let's get this equation factored and find those solutions!
The Factoring Process: A Step-by-Step Guide
Okay, let's get down to the actual factoring process for the equation 16x² + 2x - 3 = 0. This might seem a bit daunting at first, but we'll break it down into easy-to-follow steps. The key idea behind factoring is to rewrite the quadratic expression as a product of two binomials. Think of it like this: we're trying to find two expressions that look like (px + q)(rx + s) such that when we multiply them out, we get back to 16x² + 2x - 3. The first step is to look at the coefficients and the constant term. We need to find two numbers that multiply to give us the product of the leading coefficient (16) and the constant term (-3), which is 16 * -3 = -48. At the same time, these two numbers need to add up to the middle coefficient, which is 2. This might sound like a puzzle, and that's because it is! But with a bit of trial and error, you'll get the hang of it. Let’s think about pairs of factors of -48. We could have -1 and 48, -2 and 24, -3 and 16, -4 and 12, or -6 and 8. Which of these pairs adds up to 2? Bingo! -6 and 8 fit the bill. So, we've found our magic numbers: -6 and 8. Now, the next step is to rewrite the middle term (2x) using these two numbers. Instead of 2x, we'll write -6x + 8x. This gives us 16x² - 6x + 8x - 3 = 0. Notice how we haven't changed the value of the equation; we've just rewritten the middle term in a clever way.
Grouping and Factoring
Now that we've rewritten our quadratic equation as 16x² - 6x + 8x - 3 = 0, it's time to use a technique called factoring by grouping. This involves grouping the first two terms together and the last two terms together and then factoring out the greatest common factor (GCF) from each group. Let’s take the first group, 16x² - 6x. What's the greatest common factor here? Both terms are divisible by 2, and they both have at least one x. So, the GCF is 2x. Factoring out 2x from 16x² - 6x, we get 2x(8x - 3). Now let’s look at the second group, 8x - 3. Hmm, this one's a bit simpler. The greatest common factor here is just 1 (since there’s no other common factor). Factoring out 1 (which doesn't really change anything), we have 1(8x - 3). Putting these two parts together, our equation now looks like this: 2x(8x - 3) + 1(8x - 3) = 0. Do you see something interesting? Both terms now have a common factor of (8x - 3). This is exactly what we want! We can factor out this common binomial factor, just like we factor out a common number or variable. Factoring out (8x - 3), we're left with (8x - 3)(2x + 1) = 0. And there you have it! We've successfully factored our original quadratic equation into two binomial factors. This is a crucial step because it sets us up to find the solutions.
Finding the Solutions
We've done the hard work of factoring the equation 16x² + 2x - 3 = 0 into (8x - 3)(2x + 1) = 0. Now comes the fun part: finding the solutions. Remember, the solutions are the values of x that make the equation true. We're using a principle called the zero-product property. This property states that if the product of two factors is zero, then at least one of the factors must be zero. In other words, if we have A * B = 0, then either A = 0 or B = 0 (or both). This is super helpful because we've rewritten our quadratic equation as a product of two factors. So, we can set each factor equal to zero and solve for x. Let’s start with the first factor: 8x - 3 = 0. To solve for x, we add 3 to both sides of the equation, which gives us 8x = 3. Then, we divide both sides by 8 to isolate x, resulting in x = 3/8. So, one of our solutions is x = 3/8. Now, let's tackle the second factor: 2x + 1 = 0. To solve for x, we subtract 1 from both sides of the equation, giving us 2x = -1. Then, we divide both sides by 2, which gives us x = -1/2. So, our second solution is x = -1/2. We've found both solutions to the equation! The solution set is the set of all values of x that satisfy the equation. In this case, the solution set is {3/8, -1/2}. Make sure to write your solutions in the correct format as requested, usually separated by a comma and enclosed in curly braces.
Checking Our Work
Before we declare victory, it's always a good idea to check our work. This is especially important in math because a small mistake can lead to the wrong answer. Checking our solutions is pretty straightforward. We simply plug each solution back into the original equation and see if it makes the equation true. Let's start with our first solution, x = 3/8. Plugging this into the original equation, 16x² + 2x - 3 = 0, we get: 16(3/8)² + 2(3/8) - 3. Now, let's simplify. First, (3/8)² is 9/64. So, we have 16(9/64) + 2(3/8) - 3. Next, 16(9/64) simplifies to 9/4, and 2(3/8) simplifies to 3/4. So, our equation now looks like: 9/4 + 3/4 - 3. Combining the fractions, 9/4 + 3/4 equals 12/4, which simplifies to 3. So, we have 3 - 3 = 0. Yay! The equation holds true for x = 3/8. Now, let's check our second solution, x = -1/2. Plugging this into the original equation, we get: 16(-1/2)² + 2(-1/2) - 3. Simplifying, (-1/2)² is 1/4. So, we have 16(1/4) + 2(-1/2) - 3. Next, 16(1/4) simplifies to 4, and 2(-1/2) simplifies to -1. So, our equation now looks like: 4 - 1 - 3. This simplifies to 4 - 4 = 0. Double yay! The equation also holds true for x = -1/2. Since both solutions check out, we can be confident that we've solved the equation correctly. Remember, checking your work is a crucial step in problem-solving. It not only ensures accuracy but also helps solidify your understanding of the concepts.
Conclusion: Mastering Factoring
Alright, guys! We've successfully solved the quadratic equation 16x² + 2x - 3 = 0 by factoring. We found that the solution set is {-1/2, 3/8}. We walked through each step, from understanding the basic form of a quadratic equation to using the zero-product property to find the solutions. We also emphasized the importance of checking our work to ensure accuracy. Factoring quadratic equations might seem a bit tricky at first, but with practice, it becomes a powerful tool in your algebra toolkit. The key is to break down the process into manageable steps and to understand the underlying principles. Remember to look for the greatest common factor, rewrite the middle term, group the terms, and apply the zero-product property. And most importantly, always check your solutions! Factoring isn't just about getting the right answer; it's about developing your problem-solving skills and deepening your understanding of mathematical relationships. So, keep practicing, and you'll become a factoring master in no time! Keep an eye out for more math adventures, and happy solving!
