Finding The Quadratic Equation With Solutions X = (-3 ± √3 I) / 2
In the realm of mathematics, particularly within the study of quadratic equations, we often encounter solutions that extend beyond the familiar set of real numbers. These solutions, known as complex numbers, involve the imaginary unit i, which is defined as the square root of -1. Complex numbers play a crucial role in various fields, including electrical engineering, quantum mechanics, and signal processing. Understanding how to derive and interpret complex solutions of quadratic equations is a fundamental skill in mathematics.
In this article, we embark on a journey to decipher the equation that yields the complex solutions x = (-3 ± √3 i) / 2. We will delve into the quadratic formula, a powerful tool for solving quadratic equations, and explore how it leads to complex solutions when the discriminant is negative. By carefully analyzing the given solutions, we will reverse-engineer the equation that produces them, ultimately identifying the correct quadratic equation from a set of options. This exploration will not only enhance your understanding of quadratic equations and complex numbers but also sharpen your problem-solving skills in mathematics.
At the heart of solving quadratic equations lies the quadratic formula, a cornerstone of algebraic techniques. This formula provides a universal method for finding the solutions (also known as roots) of any quadratic equation in the standard form of ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The quadratic formula is expressed as:
x = (-b ± √(b² - 4ac)) / (2a)
The formula reveals that the solutions for x are determined by the coefficients a, b, and c. The expression within the square root, b² - 4ac, is known as the discriminant. The discriminant plays a pivotal role in determining the nature of the solutions. When the discriminant is positive, the equation has two distinct real solutions. When it is zero, the equation has one real solution (a repeated root). And when the discriminant is negative, as in our case, the equation has two complex solutions.
Complex solutions arise because the square root of a negative number involves the imaginary unit i, where i² = -1. This allows us to express solutions that extend beyond the real number line. The solutions we are given, x = (-3 ± √3 i) / 2, clearly indicate a negative discriminant and thus, complex solutions. Our task is to find the quadratic equation whose coefficients, when plugged into the quadratic formula, produce these specific complex solutions.
To identify the correct quadratic equation, we need to reverse-engineer the process. Starting from the given solutions, x = (-3 ± √3 i) / 2, we will work backward to reconstruct the original equation. This involves a careful comparison with the quadratic formula and some algebraic manipulation. Let's break down the solutions into their components:
x = (-3 ± √3 i) / 2 = -3/2 ± (√3 / 2) i
Comparing this with the quadratic formula, we can deduce some relationships between the coefficients a, b, and c. The term -b/(2a) corresponds to the real part of the solution, which is -3/2. The term √( b² - 4ac ) / (2a) corresponds to the imaginary part, which is (√3 / 2) i. This gives us two crucial pieces of information:
- -b/(2a) = -3/2
- √( b² - 4ac ) / (2a) = (√3 / 2) i
From the first equation, we can infer a relationship between a and b. Multiplying both sides by -2, we get b/a* = 3. This suggests that b is 3 times a. Now, let's consider the second equation. To eliminate the square root, we square both sides:
(b² - 4ac) / (4a²) = -3/4
Note that we have -3/4 on the right side because we are squaring the imaginary unit i, which gives us -1. Now, we can multiply both sides by 4a² to get:
b² - 4ac = -3a²
This equation provides another crucial link between a, b, and c. We now have two equations that relate these coefficients:
- b = 3a
- b² - 4ac = -3a²
By substituting the first equation into the second, we can eliminate b and obtain a relationship between a and c. This will help us narrow down the possible quadratic equations.
Substituting b = 3a into the equation b² - 4ac = -3a², we get:
(3a)² - 4ac = -3a²
Expanding and simplifying, we have:
9a² - 4ac = -3a²
Adding 3a² to both sides, we get:
12a² - 4ac = 0
Now, we can factor out 4a from the left side:
4a (3a - c) = 0
Since a cannot be zero (as it is the coefficient of the x² term in a quadratic equation), we must have:
3a - c = 0
This implies that c = 3a. Now we have a relationship between a, b, and c:
- b = 3a
- c = 3a
This tells us that b and c are both 3 times a. We can choose a simple value for a, such as a = 1, to find corresponding values for b and c. If a = 1, then b = 3 and c = 3. This gives us the quadratic equation:
x² + 3x + 3 = 0
However, we must also consider the possibility that the equation is a multiple of this one. To check this, we can multiply the equation by a constant. Let's multiply by 2 to get:
2x² + 6x + 6 = 0
This equation has the same solutions as x² + 3x + 3 = 0. We will now compare this result with the given options to identify the correct equation.
Now that we have derived a potential quadratic equation, we must compare it with the given options to identify the correct one. The options are:
A. x² + 3x + 3 = 0 B. 2x² + 6x + 9 = 0 C. 2x² + 6x + 3 = 0 D. x² + 3x + 12 = 0
Our derived equation, x² + 3x + 3 = 0, matches option A. However, we also considered the possibility of a multiple of this equation. Let's examine the other options.
Option B, 2x² + 6x + 9 = 0, has coefficients that are not multiples of our derived equation. Specifically, the constant term 9 is not 3 times the coefficient of x², which is 2. Therefore, this option is incorrect.
Option C, 2x² + 6x + 3 = 0, also does not match our derived equation. The constant term 3 is not 3 times the coefficient of x², which is 2. This option is also incorrect.
Option D, x² + 3x + 12 = 0, has a constant term 12, which is not 3 times the coefficient of x², which is 1. This option is incorrect as well.
Therefore, the only equation that matches our derived form is option A:
x² + 3x + 3 = 0
To ensure our answer is correct, we can use the quadratic formula to solve the equation x² + 3x + 3 = 0. In this case, a = 1, b = 3, and c = 3. Plugging these values into the quadratic formula, we get:
x = (-3 ± √(3² - 4 * 1 * 3)) / (2 * 1)
Simplifying, we have:
x = (-3 ± √(9 - 12)) / 2
x = (-3 ± √(-3)) / 2
x = (-3 ± √3 i) / 2
These are precisely the solutions given in the problem statement. This confirms that the correct equation is indeed x² + 3x + 3 = 0.
In this exploration, we successfully identified the quadratic equation that has the complex solutions x = (-3 ± √3 i) / 2. We achieved this by understanding the quadratic formula, analyzing the relationship between the coefficients and solutions, and reverse-engineering the equation. This process involved substituting values, simplifying expressions, and comparing our results with the given options.
By mastering these techniques, you can confidently tackle similar problems involving quadratic equations and complex solutions. The ability to work backward from solutions to equations is a valuable skill in mathematics and can be applied in various contexts. Remember, the quadratic formula is a powerful tool, and understanding the role of the discriminant is crucial for determining the nature of the solutions. As you continue your mathematical journey, practice and perseverance will further solidify your understanding of these concepts.