Solving 7x² + 11x² - 13x + 3 = 0: A Step-by-Step Guide

Let's dive into finding the real solutions for the equation 7x² + 11x² - 13x + 3 = 0. This equation looks like a polynomial, and we're going to use some algebraic techniques to crack it. So, buckle up, math enthusiasts! We're about to embark on a solution-finding adventure. Our main goal here is to simplify the equation, look for potential factoring opportunities, and then, if needed, deploy the trusty quadratic formula or other methods to pinpoint those elusive real solutions. Let’s get started!

Simplify and Combine Like Terms

First things first, let’s simplify this equation. We notice that we have two terms with x², namely 7x² and 11x². Combining these like terms is our initial step to make the equation more manageable. This involves adding their coefficients. So, 7x² + 11x² becomes 18x². Now, our equation looks cleaner and simpler: 18x² - 13x + 3 = 0. See how much better that looks? It’s like decluttering your room – the mathematical space feels much more inviting now!

Simplifying the equation is a crucial step because it sets the stage for further analysis. When we combine like terms, we reduce the complexity, making it easier to identify patterns and apply appropriate solution methods. Imagine trying to navigate a maze with lots of twists and turns versus one that's been streamlined – the latter is much simpler, right? Similarly, in algebra, simplification allows us to see the core structure of the equation without being distracted by unnecessary details. Now that we've got a simplified quadratic equation, we can start thinking about how to solve it. This often involves looking for ways to factor the quadratic, which can lead us directly to the roots. If factoring doesn't immediately jump out at us, we've got other tools in our arsenal, such as the quadratic formula, ready to be deployed. So, by taking this initial step of simplification, we've significantly enhanced our problem-solving capabilities.

Attempt to Factor the Quadratic

Now that we've simplified our equation to 18x² - 13x + 3 = 0, the next logical step is to see if we can factor it. Factoring, if possible, is a neat way to break down a quadratic equation into two simpler linear equations, making it much easier to find the solutions. We are looking for two binomials that, when multiplied together, give us the quadratic equation we have. Factoring involves finding two numbers that multiply to give the product of the leading coefficient (18) and the constant term (3), which is 54, and add up to the middle coefficient (-13). It's like a puzzle where we need to find the right pieces that fit together perfectly.

So, let’s think about the factors of 54. We have 1 and 54, 2 and 27, 3 and 18, 6 and 9. Now, we need to see if any of these pairs can be combined (with appropriate signs) to give us -13. After a bit of trial and error, we’ll realize that none of these pairs work directly. This doesn’t mean there aren’t solutions; it just means factoring isn't straightforward in this case. When simple factoring doesn’t work, don’t worry! This is quite common, and it's why we have other methods like the quadratic formula. The quadratic formula is our backup plan, a reliable tool that works for any quadratic equation, regardless of whether it can be easily factored. The beauty of the quadratic formula is its generality – it applies universally to all quadratic equations, providing a systematic way to find solutions. So, while factoring is a handy shortcut when it works, we're not stuck if it doesn't. We simply transition to a more robust method that guarantees we can find the solutions, if they exist. The key takeaway here is that in problem-solving, having multiple strategies and knowing when to apply each one is crucial.

Apply the Quadratic Formula

Since we couldn't easily factor the quadratic equation 18x² - 13x + 3 = 0, it's time to bring out the big guns: the quadratic formula. This formula is a powerful tool that gives us the solutions to any quadratic equation in the form ax² + bx + c = 0. The quadratic formula is x = [-b ± sqrt(b² - 4ac)] / (2a). In our equation, we can identify a = 18, b = -13, and c = 3. Plugging these values into the formula will give us the solutions for x.

Let's carefully substitute the values into the formula: x = [-(-13) ± sqrt((-13)² - 4 * 18 * 3)] / (2 * 18). This simplifies to x = [13 ± sqrt(169 - 216)] / 36. Now, let's focus on what's under the square root. We have 169 - 216, which equals -47. Uh-oh, we've encountered a negative number under the square root. This is a significant observation because it tells us something important about the solutions. Remember, the square root of a negative number is not a real number; it's an imaginary number. This means that the solutions to our equation will be complex numbers, not real numbers. So, even though the quadratic formula is a universal tool, it also provides valuable information about the nature of the solutions – whether they are real or complex. Understanding this distinction is key in mathematics because it guides us in how we interpret and use the solutions. In this case, the negative discriminant tells us that there are no real solutions, but we will have complex solutions if we want to find them.

Determine the Nature of Solutions

As we saw when applying the quadratic formula, we ended up with a negative number under the square root (√-47). This is a crucial piece of information because it tells us about the nature of the solutions to our quadratic equation. In mathematics, the expression under the square root in the quadratic formula, b² - 4ac, is called the discriminant. The discriminant is a powerful indicator of the types of solutions we'll find. If the discriminant is positive, we have two distinct real solutions. If it's zero, we have exactly one real solution (a repeated root). And, as in our case, if the discriminant is negative, we have no real solutions; instead, we have two complex solutions.

In our scenario, the discriminant (-47) is negative. This means that the solutions to the equation 18x² - 13x + 3 = 0 are complex numbers, not real numbers. Complex numbers involve the imaginary unit 'i,' which is defined as the square root of -1. So, while we can find the complex solutions if we want to delve into complex number territory, the question specifically asks for real solutions. Since there are no real numbers that satisfy this equation, our answer will reflect that. Understanding the discriminant is incredibly useful in various mathematical contexts. It allows us to quickly assess the solvability of a quadratic equation within the realm of real numbers and directs us to the appropriate methods for finding solutions, be they real or complex. In essence, it's a roadmap for solving quadratic equations, guiding us efficiently to the correct type of solution.

State the Solution Set

So, after our mathematical journey through simplification, attempting to factor, and applying the quadratic formula, we've arrived at a clear conclusion. We discovered that the discriminant (b² - 4ac) of our quadratic equation 18x² - 13x + 3 = 0 is negative. This crucial finding tells us that there are no real solutions to this equation. The solutions are complex numbers, which are a different kind of number that includes an imaginary part. However, our task was to find real solutions, and since there aren't any, we need to express this fact in our answer.

The way we denote the absence of real solutions in a solution set is by using the empty set. The empty set is a set that contains no elements. It's like an empty box or an empty room – it's a set, but it has nothing inside. The symbol for the empty set is either {} or ∅. In the context of solution sets, using the empty set signifies that there are no values of x that satisfy the equation within the set of real numbers. It’s a concise and universally understood way to communicate that there are no real solutions. Therefore, when we say the solution set is {}, we are stating clearly and mathematically that there are no real numbers that make the equation 18x² - 13x + 3 = 0 true. This completes our solution process, providing a definitive answer to the problem.

Therefore, the solution set is {}.