Solving 3x² + 36x + 81 = 0 A Step-by-Step Guide

Hey guys! Today, we're diving into the world of quadratic equations and tackling a specific problem: finding the values of x that satisfy the equation 3x² + 36x + 81 = 0. Don't worry if you feel a bit rusty on your algebra – we'll break it down step by step so it's super easy to follow. We'll explore different methods to solve this equation, ensuring you grasp the underlying concepts and can confidently tackle similar problems in the future. So, grab your thinking caps, and let's get started!

Understanding Quadratic Equations

Before we jump into solving, let's quickly recap what a quadratic equation actually is. A quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (x in our case) is 2. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to 0. These constants play a crucial role in determining the shape and position of the parabola that represents the quadratic equation when graphed. Understanding the general form helps us identify the coefficients and apply the appropriate methods for solving the equation.

In our specific equation, 3x² + 36x + 81 = 0, we can easily identify the coefficients: a = 3, b = 36, and c = 81. These values are essential for applying various solution methods, such as factoring, completing the square, or using the quadratic formula. Each method offers a unique approach to finding the roots, and choosing the most efficient one depends on the specific characteristics of the equation. For instance, if the coefficients are large or the equation appears complex, the quadratic formula might be the most straightforward approach. However, if the equation can be easily factored, that method can be quicker and more intuitive.

Key Concepts to Remember: The goal of solving a quadratic equation is to find the values of x that make the equation true. These values are also known as the roots, solutions, or zeros of the equation. A quadratic equation can have two distinct real roots, one real root (a repeated root), or two complex roots. The nature of the roots is determined by the discriminant, which is a part of the quadratic formula. The discriminant tells us whether the roots are real or complex and whether they are distinct or repeated. Recognizing these key concepts will help you approach quadratic equations with confidence and choose the right strategy for finding the solutions.

Method 1: Simplifying and Factoring

Okay, let's dive into our first method: simplifying and factoring. This method is often the quickest way to solve a quadratic equation if you can spot the factors easily. The first step is to see if we can simplify the equation. Notice that all the coefficients in our equation, 3x² + 36x + 81 = 0, are divisible by 3. This is a fantastic opportunity to make the equation easier to work with. Factoring out the common factor of 3, we get:

3(x² + 12x + 27) = 0

Now, we can divide both sides of the equation by 3, which simplifies our equation to:

x² + 12x + 27 = 0

See how much cleaner that looks? Now, let's try to factor the quadratic expression x² + 12x + 27. Factoring involves finding two numbers that multiply to give the constant term (27) and add up to give the coefficient of the x term (12). Think about it for a moment. What two numbers fit the bill?

The numbers are 3 and 9! Because 3 multiplied by 9 equals 27, and 3 plus 9 equals 12. So, we can factor the quadratic expression as follows:

(x + 3)(x + 9) = 0

Great! We've successfully factored the quadratic equation. Now, to find the values of x that make the equation true, we set each factor equal to zero:

x + 3 = 0 or x + 9 = 0

Solving these simple linear equations gives us the solutions:

x = -3 or x = -9

And there you have it! We've found the roots of the equation using the factoring method. This method is efficient when the quadratic expression can be factored easily, making it a valuable tool in your problem-solving arsenal. However, not all quadratic equations can be easily factored, so let's explore another method that works in more general cases.

Method 2: Using the Quadratic Formula

Alright, let's tackle another method for solving quadratic equations: the quadratic formula. This formula is a powerful tool that works for any quadratic equation, regardless of whether it can be factored easily or not. It might look a little intimidating at first, but once you understand it, you'll be able to solve a wide range of problems. The quadratic formula is given by:

x = (-b ± √(b² - 4ac)) / (2a)

Remember the general form of a quadratic equation: ax² + bx + c = 0? Well, the a, b, and c in the formula refer to the coefficients in our equation. In our case, for the equation 3x² + 36x + 81 = 0, we have a = 3, b = 36, and c = 81. Now, let's plug these values into the quadratic formula:

x = (-36 ± √(36² - 4 * 3 * 81)) / (2 * 3)

Time to simplify! First, let's calculate the value inside the square root:

36² - 4 * 3 * 81 = 1296 - 972 = 324

So, our equation now looks like this:

x = (-36 ± √324) / 6

The square root of 324 is 18, so we have:

x = (-36 ± 18) / 6

Now, we have two possible solutions, one with the plus sign and one with the minus sign:

For the plus sign:

x = (-36 + 18) / 6 = -18 / 6 = -3

For the minus sign:

x = (-36 - 18) / 6 = -54 / 6 = -9

And there we have it! We've found the same solutions as before: x = -3 and x = -9. The quadratic formula is a reliable method that always works, even when factoring is difficult. It's a valuable tool to have in your mathematical toolkit.

Understanding the Discriminant: Let's take a moment to talk about the discriminant, which is the part of the quadratic formula under the square root (b² - 4ac). The discriminant tells us a lot about the nature of the roots. If the discriminant is positive, as it was in our case (324), the equation has two distinct real roots. If it's zero, the equation has one real root (a repeated root). And if it's negative, the equation has two complex roots. Understanding the discriminant can help you predict the type of solutions you'll get before you even solve the equation!

The Solutions

We've successfully solved the quadratic equation 3x² + 36x + 81 = 0 using both factoring and the quadratic formula. Both methods led us to the same solutions:

x = -9 and x = -3

The question asks us to provide the solutions from the smallest to the largest. So, the solutions are:

Smaller x = -9

Larger x = -3

Therefore, the correct answer is a) x₁ = -9 and x₂ = -3. This problem demonstrates the power of different problem-solving techniques in mathematics. Knowing multiple methods allows you to choose the most efficient one for a given problem and also provides a way to double-check your answers. Factoring is great when it's straightforward, while the quadratic formula is a reliable workhorse for any quadratic equation.

Practice Makes Perfect

Solving quadratic equations can seem tricky at first, but like any skill, it gets easier with practice. The more you work through different problems, the more comfortable you'll become with the various methods and techniques. Try solving other quadratic equations using factoring, the quadratic formula, or even completing the square (another useful method we didn't cover in detail today). You can find plenty of practice problems online or in textbooks. Don't be afraid to make mistakes – that's how we learn! And remember, understanding the underlying concepts is just as important as memorizing formulas. So, keep practicing, keep exploring, and you'll become a quadratic equation whiz in no time!

By mastering quadratic equations, you'll be well-equipped to tackle more advanced mathematical concepts in the future. These equations pop up in various fields, from physics and engineering to economics and computer science. So, the effort you put in now will pay off in the long run. Keep up the great work, guys, and happy solving!