Finding The Quadratic Equation With Complex Solutions

In the realm of mathematics, quadratic equations reign supreme as fundamental tools for modeling diverse phenomena, from projectile motion to financial growth. These equations, characterized by their second-degree polynomial form, often yield solutions that reside within the familiar domain of real numbers. However, a captivating twist emerges when the discriminant, a pivotal component of the quadratic formula, takes on a negative value. This intriguing scenario ushers in the realm of complex solutions, numbers that extend beyond the real number line and incorporate the imaginary unit, denoted as 'i'.

Deciphering Complex Solutions: A Deep Dive

Complex solutions arise when the discriminant, calculated as b² - 4ac in the quadratic equation ax² + bx + c = 0, dips below zero. The negative discriminant signals that the square root operation within the quadratic formula will encounter a negative number, necessitating the introduction of the imaginary unit 'i', defined as the square root of -1. Complex solutions, therefore, take the form of a + bi, where 'a' represents the real part and 'b' represents the imaginary part.

To embark on our quest to identify the equation with the specified complex solutions, we must first delve into the quadratic formula, a cornerstone of solving quadratic equations. This formula, expressed as:

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

provides a systematic approach to determine the roots of any quadratic equation in the standard form ax² + bx + c = 0. The expression within the square root, b² - 4ac, as we've touched upon, is the discriminant, the key to unlocking the nature of the solutions.

The Quest for the Equation: A Step-by-Step Journey

Our mission is to pinpoint the equation that yields the complex solutions x = (-3 ± √3 i) / 2. To achieve this, we'll meticulously dissect each candidate equation, employing the quadratic formula to unveil its solutions. The equation that aligns perfectly with the target solutions will be crowned as the victor.

Candidate Equation 1: 2x² + 6x + 9 = 0

Let's embark on our analysis with the first contender, 2x² + 6x + 9 = 0. Here, a = 2, b = 6, and c = 9. Plugging these values into the quadratic formula, we obtain:

$$x = \frac{-6 \pm \sqrt{6^2 - 4 \cdot 2 \cdot 9}}{2 \cdot 2}$$

Simplifying the expression under the square root:

$$x = \frac{-6 \pm \sqrt{36 - 72}}{4}$$

$$x = \frac{-6 \pm \sqrt{-36}}{4}$$

Recognizing the negative discriminant, we introduce the imaginary unit:

$$x = \frac{-6 \pm 6i}{4}$$

Further simplification yields:

$$x = \frac{-3 \pm 3i}{2}$$

Behold! These solutions bear a striking resemblance to our target solutions, but the imaginary component's coefficient deviates. Thus, this equation falls short of our quest.

Candidate Equation 2: x² + 3x + 12 = 0

Next in line is the equation x² + 3x + 12 = 0. In this case, a = 1, b = 3, and c = 12. Substituting these values into the quadratic formula:

$$x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot 12}}{2 \cdot 1}$$

Simplifying the discriminant:

$$x = \frac{-3 \pm \sqrt{9 - 48}}{2}$$

$$x = \frac{-3 \pm \sqrt{-39}}{2}$$

Introducing the imaginary unit:

$$x = \frac{-3 \pm i\sqrt{39}}{2}$$

These solutions, while complex, do not perfectly align with our target solutions. The imaginary component's coefficient differs, disqualifying this equation.

Candidate Equation 3: 2x² + 6x + 3 = 0

Our third contender is 2x² + 6x + 3 = 0, where a = 2, b = 6, and c = 3. Plugging these values into the quadratic formula:

$$x = \frac{-6 \pm \sqrt{6^2 - 4 \cdot 2 \cdot 3}}{2 \cdot 2}$$

Simplifying the discriminant:

$$x = \frac{-6 \pm \sqrt{36 - 24}}{4}$$

$$x = \frac{-6 \pm \sqrt{12}}{4}$$

These solutions are real, not complex, immediately disqualifying this equation from our quest.

Candidate Equation 4: x² + 3x + 3 = 0

Our final hope rests on the equation x² + 3x + 3 = 0, where a = 1, b = 3, and c = 3. Substituting these values into the quadratic formula:

$$x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}$$

Simplifying the discriminant:

$$x = \frac{-3 \pm \sqrt{9 - 12}}{2}$$

$$x = \frac{-3 \pm \sqrt{-3}}{2}$$

Introducing the imaginary unit:

$$x = \frac{-3 \pm i\sqrt{3}}{2}$$

$$x = \frac{-3 \pm \sqrt{3}i}{2}$$

Eureka! These solutions perfectly match our target solutions. The equation x² + 3x + 3 = 0 is the victor, the equation that begets the complex solutions x = (-3 ± √3 i) / 2.

Conclusion: The Victorious Equation

In our mathematical expedition, we've successfully identified the equation x² + 3x + 3 = 0 as the one that possesses the complex solutions x = (-3 ± √3 i) / 2. This journey has underscored the pivotal role of the quadratic formula and the discriminant in unraveling the nature of quadratic equation solutions. Complex solutions, once shrouded in mystery, now stand revealed as a natural consequence of negative discriminants, expanding the realm of solutions beyond the familiar real number line. Understanding complex numbers is crucial in various fields, including electrical engineering, quantum mechanics, and applied mathematics. This exploration reinforces the power of mathematical tools in deciphering the intricacies of equations and their solutions, paving the way for deeper insights into the mathematical universe.

This exercise not only answers a specific question but also highlights the importance of understanding the underlying principles of quadratic equations and complex numbers. By systematically applying the quadratic formula and analyzing the discriminant, we can confidently navigate the world of complex solutions and unlock the secrets they hold. The ability to identify quadratic equations with complex roots is a valuable skill in advanced mathematics and related fields.

In summary, the correct equation is x² + 3x + 3 = 0, which demonstrates how the quadratic formula can be used to find solutions, including complex ones, to quadratic equations. The process involved careful application of the formula, simplification, and recognition of the properties of complex numbers. This problem serves as a great example of how mathematical problem-solving involves a combination of algebraic techniques and conceptual understanding.