Solving Cubic Equations: Finding Real & Complex Solutions

Hey math enthusiasts! Let's dive into the fascinating world of cubic equations, specifically focusing on finding the real and complex solutions for the equation: x³ + 2x² + 36x = -72. This kind of problem often pops up in algebra, and understanding how to solve it is a fundamental skill. We'll break it down step-by-step, making sure you grasp the concepts and can tackle similar problems with confidence. The goal is to identify all the values of x that satisfy the equation, including both real and complex numbers. Complex numbers, as you might remember, involve the imaginary unit i, where i² = -1. So, buckle up, and let's get started!

Rearranging and Setting Up the Equation

The first crucial step in solving any equation is to get it into a standard form. For cubic equations, this usually means setting the equation equal to zero. This allows us to apply various solving techniques. Let's take our equation, x³ + 2x² + 36x = -72, and rearrange it. We do this by adding 72 to both sides of the equation. This gives us:

x³ + 2x² + 36x + 72 = 0

Now, the equation is in the standard form we need. This form is essential because it allows us to identify possible solutions more easily. We're essentially looking for the roots or zeros of this cubic polynomial. These are the x values where the graph of the equation crosses the x-axis. Since this is a cubic equation, we expect to find up to three solutions. These solutions can be real numbers, complex numbers, or a combination of both. In the next section, we'll begin to explore the methods for finding these solutions. Remember, the rearrangement is the cornerstone of our problem-solving strategy, and it sets the stage for the rest of the process. It's like preparing the canvas before you start painting; it's essential for a successful outcome.

Factoring by Grouping

One of the initial strategies to try when faced with a cubic equation is factoring. Factoring can simplify the equation, making it easier to identify the roots. In this case, we can try factoring by grouping. Factoring by grouping involves looking for common factors among pairs of terms. Let's revisit our equation: x³ + 2x² + 36x + 72 = 0.

We'll group the first two terms and the last two terms together:

(x³ + 2x²) + (36x + 72) = 0

Now, let's factor out the greatest common factor (GCF) from each group. From the first group (x³ + 2x²), the GCF is x². Factoring this out gives us x²(x + 2). From the second group (36x + 72), the GCF is 36. Factoring this out gives us 36(x + 2). So, our equation now looks like:

x²(x + 2) + 36(x + 2) = 0

Notice that we now have a common factor of (x + 2) in both terms. We can factor this out as well:

(x + 2)(x² + 36) = 0

This factored form is a breakthrough. It simplifies the equation significantly, turning it into a product of two factors that equals zero. This allows us to use the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This leads us to the next section where we identify the solutions.

Finding the Solutions: Real and Complex

With our equation factored as (x + 2)(x² + 36) = 0, we're ready to find the solutions. According to the zero-product property, either (x + 2) = 0 or (x² + 36) = 0. Let's solve each of these equations to find the values of x.

First, consider (x + 2) = 0. Subtracting 2 from both sides gives us:

x = -2

So, x = -2 is one of our solutions; it's a real number. Now, let's look at the second equation, (x² + 36) = 0. To solve for x, we first subtract 36 from both sides:

x² = -36

Now, to solve for x, we take the square root of both sides. Remember that the square root of a negative number involves the imaginary unit i, where i² = -1:

x = ±√(-36) x = ±6i

This gives us two complex solutions: x = 6i and x = -6i. So, we have found three solutions in total: one real solution x = -2, and two complex solutions, x = 6i and x = -6i. These solutions represent the points where the cubic equation intersects the x-axis (for the real solution) and the complex plane (for the complex solutions). The fact that we have one real and two complex solutions is perfectly valid for a cubic equation, as it can have up to three roots, which can be real or complex. In our case, the answers are -2, 6i, and -6i.

Matching with the Options

Now that we've diligently solved the cubic equation and found its solutions, let's see which of the provided options matches our results. We found that the solutions are x = -2, x = 6i, and x = -6i. Let's examine the multiple-choice options:

A. 2.8, 8.4i, -8.4i B. -2, 6i, -6i C. 1.7, 1.7i, -5.1i D. 2, -6i

By comparing the solutions we computed with the answer choices, we see that option B perfectly matches our findings. Option B lists the solutions as -2, 6i, -6i. This confirms that we have correctly solved the cubic equation and identified all the real and complex solutions. The other options do not match our solutions, indicating potential errors in those calculations. So, the correct answer is B. Our problem is complete; we have successfully navigated through the cubic equation, found the roots, and matched them with the correct answer from the given choices. This process highlights the importance of methodical steps in solving complex problems. Good work!