Solving 1.5x² + X - 8.2 = 0 Approximate Solutions Guide

Hey guys! Let's dive into solving this quadratic equation. We've got 1.5x² + x - 8.2 = 0, and we need to find the approximate solutions, rounded to the nearest hundredth. Now, this might seem a bit intimidating at first, but don't worry, we'll break it down step by step. We're going to use the quadratic formula, a trusty tool in our mathematical arsenal, to tackle this problem. Think of it as our secret weapon for solving equations like these. The quadratic formula is derived from the method of completing the square, a technique used to rewrite a quadratic equation in a form that allows for easy solution. By completing the square on the general quadratic equation ax² + bx + c = 0, we arrive at the quadratic formula, which provides a direct way to find the roots of any quadratic equation. This formula is incredibly versatile because it works for any quadratic equation, regardless of whether the roots are real or complex. The process of completing the square involves manipulating the equation to create a perfect square trinomial, which can then be factored and solved. This method not only leads to the quadratic formula but also provides a deeper understanding of the structure of quadratic equations and their solutions. The quadratic formula is a cornerstone of algebra and is essential for solving a wide range of problems in mathematics, physics, and engineering. Its derivation from completing the square highlights the power of algebraic manipulation and the elegance of mathematical reasoning. Understanding the origins and applications of the quadratic formula is crucial for anyone seeking to master quadratic equations and their solutions. So, buckle up, and let's get started!

Understanding the Quadratic Formula

The quadratic formula is your best friend when dealing with equations in the form ax² + bx + c = 0. Remember this equation? Our equation, 1.5x² + x - 8.2 = 0, fits this perfectly! So, what exactly is the quadratic formula? It's this beauty:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Where:

• a is the coefficient of x² (in our case, 1.5)
• b is the coefficient of x (in our case, 1)
• c is the constant term (in our case, -8.2)

This formula might look a little scary at first, but trust me, it's quite straightforward once you get the hang of it. It's essentially a plug-and-chug situation. You identify your a, b, and c values, pop them into the formula, and then do the arithmetic. The magic of the quadratic formula lies in its ability to handle any quadratic equation, regardless of whether the solutions are real or complex. The part under the square root, b² - 4ac, known as the discriminant, plays a crucial role in determining the nature of the roots. If the discriminant is positive, we have two distinct real roots; if it's zero, we have one real root (a repeated root); and if it's negative, we have two complex roots. This versatility makes the quadratic formula an indispensable tool for mathematicians, scientists, and engineers alike. The formula itself is derived from the method of completing the square, a fundamental technique in algebra. Understanding the derivation of the quadratic formula provides a deeper appreciation for its structure and application. It's not just a formula to memorize; it's a powerful result that stems from sound mathematical principles. So, whether you're solving a textbook problem or tackling a real-world application, the quadratic formula is your go-to solution for quadratic equations. Let's move on and see how we can apply this formula to our specific equation. We'll carefully substitute the values of a, b, and c and then simplify to find our approximate solutions. Remember, practice makes perfect, so the more you use the quadratic formula, the more comfortable you'll become with it. So, keep practicing, and you'll be solving quadratic equations like a pro in no time! Don't be intimidated by the symbols; embrace the formula and let it guide you to the solutions.

Plugging in the Values

Alright, let's get our hands dirty! We know a = 1.5, b = 1, and c = -8.2. Now, we're going to carefully substitute these values into the quadratic formula. This is where attention to detail is key. A small mistake in the substitution can lead to a completely different answer. So, let's take our time and make sure everything is in its rightful place.

$$x = \frac{-1 \pm \sqrt{1^2 - 4(1.5)(-8.2)}}{2(1.5)}$$

See how we just replaced the a, b, and c in the formula with their respective values? The negative sign in front of the 1 is crucial, so don't forget it! Also, make sure you're squaring the correct value and that you're multiplying all the terms under the square root accurately. Once you've double-checked your substitution, you're ready for the next step: simplifying the expression. Now, let's take a closer look at what's happening under the square root. We have 1² - 4(1.5)(-8.2). This is where the order of operations (PEMDAS/BODMAS) comes into play. We need to perform the multiplication before the subtraction. So, 4 * 1.5 * 8.2 gives us a positive number because we have two negative signs multiplying each other. This positive value is then added to 1², which is simply 1. The result is a positive number under the square root, indicating that we will have two distinct real solutions to our quadratic equation. This is great news because it means we're on the right track to finding our approximate solutions. Remember, if the value under the square root were negative, we would have complex solutions, which involve imaginary numbers. But in this case, we're dealing with real numbers, which makes the problem a bit simpler. So, let's continue with the simplification process. We'll calculate the value under the square root, then take the square root, and finally, we'll perform the addition and subtraction in the numerator. Don't rush; take it one step at a time, and you'll get there. With the values plugged in, the next step is to simplify. Let's do it!

Simplifying the Expression

Okay, let's simplify this beast! First, let's tackle the part under the square root:

$$1^2 - 4(1.5)(-8.2) = 1 + 49.2 = 50.2$$

So, our equation now looks like this:

$$x = \frac{-1 \pm \sqrt{50.2}}{3}$$

Now, we need to find the square root of 50.2. A calculator will be super helpful here. The square root of 50.2 is approximately 7.085.

So, we have:

$$x = \frac{-1 \pm 7.085}{3}$$

Now we're getting somewhere! We're almost at the finish line. We've simplified the expression under the square root, and we've calculated the square root itself. Now, we have two separate calculations to perform, one with the plus sign and one with the minus sign. This is because the ± symbol in the quadratic formula represents two distinct solutions to the quadratic equation. One solution is obtained by adding the square root term, and the other solution is obtained by subtracting it. This is a fundamental aspect of solving quadratic equations, and it's important to keep track of both possibilities. So, let's separate these two cases and calculate each solution individually. We'll first handle the case with the plus sign, adding 7.085 to -1 and then dividing the result by 3. This will give us one of our approximate solutions. Then, we'll handle the case with the minus sign, subtracting 7.085 from -1 and then dividing the result by 3. This will give us our second approximate solution. By calculating both solutions, we'll have a complete picture of the roots of our quadratic equation. Remember, quadratic equations can have up to two distinct solutions, so it's crucial to consider both possibilities. So, let's proceed with the calculations, carefully handling the addition and subtraction, and we'll soon have our approximate solutions rounded to the nearest hundredth. We're almost there; let's keep going!

Calculating the Two Solutions

We have two scenarios to consider:

1. Using the plus sign:

   $$x = \frac{-1 + 7.085}{3} = \frac{6.085}{3} \approx 2.028$$

2. Using the minus sign:

   $$x = \frac{-1 - 7.085}{3} = \frac{-8.085}{3} \approx -2.695$$

So, we have two approximate solutions: 2.028 and -2.695. But wait, we need to round these to the nearest hundredth! Rounding is a crucial step in many mathematical problems, especially when dealing with approximate solutions. It's important to understand the rules of rounding and to apply them consistently. In this case, we need to round our solutions to two decimal places, which means we'll be looking at the digit in the thousandths place to determine whether to round up or down. For the first solution, 2.028, the digit in the thousandths place is 8, which is greater than or equal to 5, so we round up the hundredths digit. This gives us an approximate solution of 2.03. For the second solution, -2.695, the digit in the thousandths place is 5, which means we also round up the hundredths digit. However, since we're dealing with a negative number, rounding up actually means making the number more negative. So, -2.695 rounded to the nearest hundredth is -2.70. It's essential to pay attention to the sign of the number when rounding, as it can affect the direction of the rounding. Now that we've rounded our solutions to the nearest hundredth, we have our final approximate solutions to the quadratic equation. These values are the points where the parabola represented by the equation intersects the x-axis. They are also known as the roots or zeros of the equation. So, we've successfully navigated the quadratic formula, simplified the expression, calculated the two solutions, and rounded them appropriately. We're done! Now, let's present our final answer.

Final Answer

Rounding to the nearest hundredth, our approximate solutions are:

• x ≈ 2.03
• x ≈ -2.70

And there you have it! We've successfully found the approximate solutions to the equation 1.5x² + x - 8.2 = 0. Give yourselves a pat on the back! Solving quadratic equations can be a bit of a rollercoaster, but with the quadratic formula in your toolkit, you're well-equipped to tackle these problems. Remember, the key is to understand the formula, plug in the values carefully, simplify step by step, and don't forget to round your answers appropriately. Practice makes perfect, so the more you work with quadratic equations, the more confident you'll become in solving them. The quadratic formula is not just a mathematical tool; it's a gateway to understanding a wide range of problems in various fields, from physics to engineering to finance. Quadratic equations pop up in all sorts of real-world applications, so mastering the art of solving them is a valuable skill to have. Whether you're calculating the trajectory of a projectile, designing a bridge, or analyzing financial data, quadratic equations can provide insights and solutions. So, keep honing your skills, keep practicing, and you'll be amazed at the power of quadratic equations and the quadratic formula. And remember, if you ever get stuck, don't hesitate to review the steps, ask for help, or try a different approach. Math is a journey of exploration and discovery, and every problem is an opportunity to learn and grow. So, embrace the challenges, celebrate the successes, and keep exploring the fascinating world of mathematics!