Finding Quadratic Equation With Complex Solutions
Hey guys! Let's tackle a fascinating problem today that involves complex numbers and quadratic equations. We're on the hunt for the equation that spits out these solutions: $x=\frac{-3 \pm \sqrt{3} i}{2}$. It looks a bit intimidating at first, but trust me, we'll break it down step-by-step and make it super clear. We've got four potential quadratic equations lined up, and our mission is to find the one that perfectly matches those solutions. This isn't just about plugging numbers in; it's about understanding the relationship between roots and coefficients, and how complex numbers play their part in the world of quadratics. So, grab your thinking caps, and let's dive in!
Decoding the Complex Roots
First, let’s really understand what these complex roots $x=\frac{-3 \pm \sqrt{3} i}{2}$ are telling us. The presence of i (the imaginary unit, where i² = -1) immediately clues us in that we're dealing with a quadratic equation that has no real roots. This happens when the discriminant (the part under the square root in the quadratic formula) is negative. These roots are in the form of a + bi and a - bi, where a is the real part and b is the imaginary part. In our case, a = -3/2 and b = ±√3/2. Knowing this is crucial because the structure of the roots directly relates to the coefficients of the quadratic equation. Think about it: the quadratic formula itself reveals how the coefficients (the numbers in front of the x² term, the x term, and the constant term) dictate the roots. So, by carefully examining our roots, we can work backward to reconstruct the equation. We'll explore how the sum and product of these roots can guide us directly to the correct coefficients. This is a classic technique in algebra, and mastering it will make solving these types of problems a breeze. We need to find the quadratic equation that embodies these specific roots, and we’re going to do it by unraveling the connection between roots and equations.
The Sum and Product of Roots: Our Secret Weapons
Now, let’s bring in our secret weapons: the sum and product of roots. For any quadratic equation in the standard form ax² + bx + c = 0, there's a beautiful relationship between the roots (let's call them r₁ and r₂) and the coefficients: the sum of the roots (r₁ + r₂) is equal to -b/ a, and the product of the roots (r₁ r₂) is equal to c/ a. This is a fundamental concept that makes solving these problems much more manageable. So, let’s calculate the sum and product of our given roots. The sum is ((-3 + √3 i) / 2) + ((-3 - √3 i) / 2). Notice how the imaginary parts neatly cancel each other out, leaving us with a real number. This is a characteristic of complex conjugate pairs – their sum is always real. Then, we calculate the product: ((-3 + √3 i) / 2) * ((-3 - √3 i) / 2). This is where things get interesting. We're multiplying two complex conjugates, which will result in a real number (remember (a + bi)(a - bi) = a² + b²). After doing the math, we'll have the sum and product, and these values are the golden keys to unlocking our equation. We can then compare these calculated values with the ratios of the coefficients in our four potential equations. This method is not just efficient; it also deepens our understanding of the structure of quadratic equations and their solutions. By finding the sum and the product, we’re essentially reverse-engineering the quadratic formula, giving us a powerful shortcut to the answer.
Cracking the Code: Calculating Sum and Product
Okay, let's get our hands dirty and calculate the sum and product of the roots. Remember, our roots are $x=\frac-3 + \sqrt{3} i}{2}$ and $x=\frac{-3 - \sqrt{3} i}{2}$. First, the sum i}2}) + (\frac{-3 - \sqrt{3} i}{2}) = \frac{-3 + \sqrt{3} i - 3 - \sqrt{3} i}{2}$. Notice the magic – the imaginary terms (+√3 i and -√3 i) cancel each other out! This is exactly what we expect when adding complex conjugates. So, we're left with (-3 - 3) / 2 = -6 / 2 = -3. Therefore, the sum of the roots is -3. Now, for the product i}2}) * (\frac{-3 - \sqrt{3} i}{2}) = \frac{(-3)^2 - (\sqrt{3} i)^2}{2 * 2} = \frac{9 - (3 * i^2)}{4}$. Remember that i² = -1, so we can simplify further{4} = \frac{9 + 3}{4} = \frac{12}{4} = 3$. So, the product of the roots is 3. We now have both the sum (-3) and the product (3). These are our key pieces of information. Next, we’ll use these values to identify the correct quadratic equation from our options.
The Coefficient Connection: Matching Sum and Product
Now comes the exciting part: connecting the sum and product of roots we just calculated to the coefficients of our potential quadratic equations. Remember, for a quadratic equation ax² + bx + c = 0, the sum of the roots is -b/ a and the product of the roots is c/ a. We found that the sum of our roots is -3, and the product is 3. So, we need to find an equation where -b/ a = -3 and c/ a = 3. Let's examine our options one by one. This is where careful observation and a bit of algebraic intuition come into play. We're looking for an equation where the ratio of the coefficients matches our sum and product values. It's like fitting puzzle pieces together – the numbers must align perfectly. By systematically checking each equation, we'll narrow down the possibilities and pinpoint the one that satisfies both conditions. This process reinforces the link between the roots and coefficients, giving us a deeper appreciation for the structure of quadratic equations. It's not just about memorizing formulas; it's about understanding the underlying relationships and using them to solve problems effectively.
The Final Showdown: Identifying the Winning Equation
Let's put our detective hats on and go through each of the given equations to see which one fits our criteria. We need an equation where -b/ a = -3 and c/ a = 3.
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2x² + 6x + 9 = 0: Here, a = 2, b = 6, and c = 9. So, -b/ a = -6 / 2 = -3, which matches our sum. But, c/ a = 9 / 2, which does not match our product of 3. So, this equation is out.
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x² + 3x + 12 = 0**: In this case, a = 1, b = 3, and c = 12. Then, -b/ a = -3 / 1 = -3 (sum matches!). However, c/ a = 12 / 1 = 12, which is way off from our product of 3. So, this one doesn't work either.
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x² + 3x + 3 = 0**: Here, a = 1, b = 3, and c = 3. We have -b/ a = -3 / 1 = -3 (sum matches!) and c/ a = 3 / 1 = 3 (product matches!). Bingo! This equation satisfies both conditions.
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2x² + 6x + 3 = 0: For this equation, a = 2, b = 6, and c = 3. So, -b/ a = -6 / 2 = -3 (sum matches!), but c/ a = 3 / 2, which doesn't match our product of 3. So, this one is not the winner.
It's clear that the equation x² + 3x + 3 = 0** is the one we've been searching for. It's like finding the last piece of a puzzle, and it feels incredibly satisfying when everything clicks into place.
Victory Lap: The Equation Revealed
Alright guys, we did it! After carefully analyzing the complex roots, calculating their sum and product, and comparing them to the coefficients of the given equations, we've pinpointed the correct answer. The equation that has the solutions $x=\frac{-3 \pm \sqrt{3} i}{2}$ is x² + 3x + 3 = 0**. This wasn't just about blindly applying formulas; it was about understanding the deep connection between the roots and the coefficients of a quadratic equation. We used the power of the sum and product of roots, a fantastic tool that simplifies these kinds of problems. By working through this step-by-step, we’ve not only found the solution but also reinforced our understanding of quadratic equations and complex numbers. Remember, in mathematics, the journey is just as important as the destination. We've honed our problem-solving skills, strengthened our algebraic intuition, and gained a deeper appreciation for the elegance of mathematical relationships. So, pat yourselves on the back, and let's keep exploring the fascinating world of mathematics!
