Complex Solutions: Solving 3x² - 3x + 7 = 0
Hey Plastik Magazine readers! Let's dive into the fascinating world of complex numbers and solve a quadratic equation that might seem a bit tricky at first glance. We're tackling the equation 3x² - 3x + 7 = 0, and our goal is to find all its complex solutions. So, grab your thinking caps, and let's get started!
Understanding Quadratic Equations and Complex Numbers
Before we jump into solving our specific equation, let's refresh some key concepts. Quadratic equations are polynomial equations of the second degree, generally written in the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to a quadratic equation are also called its roots, and these roots can be real or complex numbers.
Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1 (i² = -1). The real part of the complex number is a, and the imaginary part is b. Complex numbers are crucial because they allow us to find solutions to equations that have no real roots. This often happens when the discriminant of a quadratic equation is negative, which we'll see in our example.
When dealing with quadratic equations, one of the most important tools we have is the quadratic formula. This formula provides a direct way to find the solutions of any quadratic equation in the form ax² + bx + c = 0. The quadratic formula is given by:
x = (-b ± √(b² - 4ac)) / (2a)
Inside the square root, we have the discriminant (b² - 4ac). The discriminant tells us a lot about the nature of the roots:
- If b² - 4ac > 0, the equation has two distinct real roots.
- If b² - 4ac = 0, the equation has one real root (a repeated root).
- If b² - 4ac < 0, the equation has two complex roots.
So, why are complex numbers so important? Well, they allow us to solve any quadratic equation, regardless of whether it has real solutions or not. Without complex numbers, we'd be stuck when trying to solve equations where the discriminant is negative. Complex numbers pop up in various fields, including electrical engineering, quantum mechanics, and signal processing. They’re not just some abstract mathematical concept; they have real-world applications!
Applying the Quadratic Formula to Our Equation
Alright, let's get back to our equation: 3x² - 3x + 7 = 0. To find the complex solutions, we'll use the quadratic formula. First, we need to identify the coefficients a, b, and c:
- a = 3
- b = -3
- c = 7
Now, we'll plug these values into the quadratic formula:
x = (-(-3) ± √((-3)² - 4 * 3 * 7)) / (2 * 3)
Let's simplify this step by step. First, we deal with the negative signs and the multiplication:
x = (3 ± √(9 - 84)) / 6
Next, we simplify the expression inside the square root:
x = (3 ± √(-75)) / 6
Uh-oh, we have a negative number under the square root! This tells us that the solutions will be complex numbers. Let's continue simplifying. We can rewrite √(-75) using the imaginary unit i:
√(-75) = √(75 * -1) = √(75) * √(-1) = √(25 * 3) * i = 5√3 * i
Now, substitute this back into our equation:
x = (3 ± 5√3 * i) / 6
Finally, we can simplify by dividing both the real and imaginary parts by 6:
x = (3/6) ± (5√3 * i / 6)
x = (1/2) ± (5√3 / 6)i
So, we have two complex solutions:
- x₁ = (1/2) + (5√3 / 6)i
- x₂ = (1/2) - (5√3 / 6)i
These are the complex roots of the equation 3x² - 3x + 7 = 0. Notice that they are complex conjugates of each other, meaning they have the same real part but opposite imaginary parts. This is a common occurrence when dealing with quadratic equations that have a negative discriminant.
Verifying the Solutions
To make sure we've got the correct solutions, it's always a good idea to verify them. We can do this by plugging each solution back into the original equation and checking if it equals zero. Let's start with the first solution, x₁ = (1/2) + (5√3 / 6)i:
3x₁² - 3x₁ + 7 = 3[(1/2) + (5√3 / 6)i]² - 3[(1/2) + (5√3 / 6)i] + 7
First, we need to square the complex number:
[(1/2) + (5√3 / 6)i]² = (1/4) + (5√3 / 6)i - (75/36) = (1/4) - (75/36) + (5√3 / 6)i
Simplify the real part:
(1/4) - (75/36) = (9/36) - (75/36) = -66/36 = -11/6
So, the square is:
(-11/6) + (5√3 / 6)i
Now, multiply by 3:
3[(-11/6) + (5√3 / 6)i] = -11/2 + (5√3 / 2)i
Next, we subtract 3x₁:
-11/2 + (5√3 / 2)i - 3[(1/2) + (5√3 / 6)i] = -11/2 + (5√3 / 2)i - 3/2 - (5√3 / 2)i
The imaginary parts cancel out, and we have:
-11/2 - 3/2 = -14/2 = -7
Finally, add 7:
-7 + 7 = 0
Great! The first solution checks out. Now, let's quickly verify the second solution, x₂ = (1/2) - (5√3 / 6)i. We'll follow a similar process:
3x₂² - 3x₂ + 7 = 3[(1/2) - (5√3 / 6)i]² - 3[(1/2) - (5√3 / 6)i] + 7
Squaring the complex number:
[(1/2) - (5√3 / 6)i]² = (1/4) - (5√3 / 6)i - (75/36) = -11/6 - (5√3 / 6)i
Multiply by 3:
3[(-11/6) - (5√3 / 6)i] = -11/2 - (5√3 / 2)i
Subtract 3x₂:
-11/2 - (5√3 / 2)i - 3[(1/2) - (5√3 / 6)i] = -11/2 - (5√3 / 2)i - 3/2 + (5√3 / 2)i
Again, the imaginary parts cancel out, and we have:
-11/2 - 3/2 = -14/2 = -7
Add 7:
-7 + 7 = 0
Excellent! Both solutions satisfy the original equation. We've successfully found and verified the complex solutions.
Visualizing Complex Solutions
Complex numbers can be visualized on a complex plane, which is similar to the Cartesian plane but with the horizontal axis representing the real part and the vertical axis representing the imaginary part. Our solutions, x₁ = (1/2) + (5√3 / 6)i and x₂ = (1/2) - (5√3 / 6)i, can be plotted as points on this plane.
The real part of both solutions is 1/2, so both points will lie on the vertical line Re(z) = 1/2. The imaginary parts are (5√3 / 6) and -(5√3 / 6), which are approximately 1.44 and -1.44, respectively. Thus, x₁ is located in the first quadrant, and x₂ is located in the fourth quadrant. The two points are reflections of each other across the real axis, which is another characteristic of complex conjugate pairs.
Visualizing complex numbers helps in understanding their properties and operations. It’s a powerful way to grasp the concept that these numbers are not just abstract entities but have a geometric interpretation.
Conclusion
So there you have it, guys! We've successfully found the complex solutions to the quadratic equation 3x² - 3x + 7 = 0 using the quadratic formula. We identified the coefficients, plugged them into the formula, simplified the result, and verified our solutions. We also touched on the importance of complex numbers and how they enable us to solve equations that have no real roots. Plus, we had a peek at how these solutions can be visualized on the complex plane.
Remember, complex numbers might seem a bit intimidating at first, but they're an essential part of mathematics and have many practical applications. Keep practicing, and you'll become a pro at solving complex equations in no time! Stay tuned for more math adventures here at Plastik Magazine!