Solving 0 = X² - 5x - 4: Find The Negative Solution

Introduction: Understanding Quadratic Equations and the Quadratic Formula

Hey guys! Let's dive into the fascinating world of quadratic equations, those mathematical expressions that often look like a rollercoaster ride on a graph. A quadratic equation is basically any equation that can be written in the form ax² + bx + c = 0, where a, b, and c are constants, and a isn't zero. These equations pop up everywhere, from physics problems describing projectile motion to engineering designs for bridges and arches. Solving them is a crucial skill in mathematics, and one of the most reliable tools we have is the quadratic formula. This formula is a magical key that unlocks the solutions (also called roots or zeros) of any quadratic equation, no matter how complex it might seem. The quadratic formula is given by: x = (-b ± √(b² - 4ac)) / 2a. It might look a bit intimidating at first, but trust me, once you get the hang of it, it's like riding a bike – you'll never forget! The beauty of this formula lies in its ability to handle any quadratic equation, whether the solutions are real numbers (positive, negative, or zero) or complex numbers (involving the imaginary unit i). It’s a one-stop-shop for finding those elusive x values that make the equation true. In this article, we're going to help Joline solve a specific quadratic equation using this very formula, focusing on finding the negative real number solution. So, buckle up, and let's embark on this mathematical adventure together!

Joline's Challenge: Deconstructing the Equation 0 = x² - 5x - 4

Our friend Joline is tackling the quadratic equation 0 = x² - 5x - 4. This equation perfectly fits the standard form ax² + bx + c = 0, and our mission is to find the values of x that make this equation a true statement. To successfully apply the quadratic formula, the first step is to identify the coefficients a, b, and c. In Joline's equation, we can see that: a is the coefficient of x², which is 1 (since x² is the same as 1x²). b is the coefficient of x, which is -5. c is the constant term, which is -4. Understanding these coefficients is like knowing the ingredients of a recipe – you need them to bake the cake (or, in this case, solve the equation!). Once we've correctly identified a, b, and c, we can confidently plug them into the quadratic formula. This careful preparation is key to avoiding errors and ensuring we arrive at the correct solutions. Now, with our ingredients ready, let's dive into the heart of the matter: applying the quadratic formula to find those x values. Remember, the goal is to find the specific value of x that is a negative real number. So, we'll be paying close attention to the signs and the nature of the solutions we obtain. This methodical approach will lead us to the answer, just like following a map leads to the treasure!

Applying the Quadratic Formula: A Step-by-Step Guide

Now comes the exciting part – using the quadratic formula to solve Joline’s equation, 0 = x² - 5x - 4. Remember the formula? It's x = (-b ± √(b² - 4ac)) / 2a. We've already identified our coefficients: a = 1, b = -5, and c = -4. Let’s carefully substitute these values into the formula. First, we replace b with -5, a with 1, and c with -4. This gives us: x = (-(-5) ± √((-5)² - 4 * 1 * -4)) / (2 * 1). Notice how we've used parentheses to maintain the correct signs – a crucial step to avoid common mistakes. Next, we simplify the expression step-by-step. The double negative in -(-5) becomes +5. Inside the square root, (-5)² is 25, and -4 * 1 * -4 is +16. So, the expression under the square root becomes 25 + 16 = 41. The denominator simplifies to 2 * 1 = 2. Now our equation looks much cleaner: x = (5 ± √41) / 2. This tells us that we have two possible solutions for x, one with a plus sign and one with a minus sign. These are x = (5 + √41) / 2 and x = (5 - √41) / 2. We’re getting closer to finding the negative real solution, so let’s move on to the next step: evaluating these expressions and identifying the solution we need.

Finding the Solutions: Identifying the Negative Real Root

We've arrived at the point where we have two potential solutions for x: x = (5 + √41) / 2 and x = (5 - √41) / 2. Now, we need to determine which of these is the negative real number solution that Joline is looking for. To do this, we'll evaluate each expression. Let's start with x = (5 + √41) / 2. The square root of 41 is approximately 6.4. So, this expression becomes approximately (5 + 6.4) / 2 = 11.4 / 2 = 5.7. This solution is positive, so it's not the one we're after. Now let’s look at the second solution: x = (5 - √41) / 2. Using our approximation of √41 as 6.4, this becomes (5 - 6.4) / 2 = -1.4 / 2 = -0.7. This solution is negative! Therefore, x = (5 - √41) / 2 is the negative real number solution to Joline's equation. To be precise, we were asked to round to the nearest tenth if necessary. Our approximate calculation gives us -0.7. To ensure accuracy, we can use a calculator to find a more precise value for √41, which is approximately 6.403. Plugging this in, we get (5 - 6.403) / 2 = -1.403 / 2 = -0.7015. Rounding this to the nearest tenth, we still get -0.7. So, we've successfully found the solution Joline was seeking! In the next section, we'll wrap up our findings and highlight the key steps we took.

Conclusion: Joline's Solution and Key Takeaways

Alright, guys! We've successfully navigated the world of quadratic equations and helped Joline find the negative real number solution to her equation, 0 = x² - 5x - 4. By carefully applying the quadratic formula, x = (-b ± √(b² - 4ac)) / 2a, and identifying the coefficients a = 1, b = -5, and c = -4, we were able to break down the problem into manageable steps. We calculated the two possible solutions: x = (5 + √41) / 2 and x = (5 - √41) / 2. After evaluating these expressions, we determined that x = (5 - √41) / 2 is the negative real solution. Rounding to the nearest tenth, we found the solution to be approximately -0.7. This exercise highlights the power and versatility of the quadratic formula as a tool for solving a wide range of equations. It also underscores the importance of careful substitution, simplification, and attention to signs to avoid errors. Remember, practice makes perfect! The more you work with quadratic equations and the quadratic formula, the more comfortable and confident you'll become. So, keep exploring, keep solving, and keep having fun with math! You've got this!