Which Of The Following Equations Have No Real Solutions But Two Complex Solutions? \( \begin{tabular}{|c|c|} \hline $3x^2 - 5x = -8$ & $2x^2 = 6x - 5$ \\ \hline $12x = 9x^2 + 4$ & $-x^2 - 10x = 34$\\ \hline \end{tabular} \)

In the realm of mathematics, understanding the nature of solutions to quadratic equations is a fundamental concept. When we delve into quadratic equations, we often encounter scenarios where the solutions aren't real numbers but fall into the realm of complex numbers. This article aims to explore equations that exhibit this unique characteristic – equations that have no real solutions but possess two complex solutions. We will dissect the criteria that lead to such solutions and analyze several examples to solidify our understanding. To truly grasp this concept, we need to revisit the quadratic formula and the discriminant, which act as our guiding tools in determining the nature of roots. This journey into complex solutions is not just an abstract mathematical exercise; it has profound implications in various fields, including engineering, physics, and computer science. The ability to identify and work with complex solutions broadens our mathematical toolkit and allows us to tackle problems that are unsolvable within the confines of real numbers. So, let's embark on this exploration, armed with curiosity and a desire to unravel the mysteries of quadratic equations and their complex solutions.

The Discriminant: Unveiling the Nature of Roots

The discriminant, a key component of the quadratic formula, plays a pivotal role in determining the nature of the roots of a quadratic equation. The quadratic formula, given by ${ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} }$, where ${ a }$, ${ b }$, and ${ c }$ are coefficients of the quadratic equation ${ ax^2 + bx + c = 0 }$, contains a term under the square root, ${ b^2 - 4ac }$. This term is the discriminant, often denoted as ${ \Delta }$. The value of the discriminant dictates whether the roots are real, complex, or repeated. When the discriminant is positive (${ \Delta > 0 }$), the quadratic equation has two distinct real roots. This scenario corresponds to the parabola intersecting the x-axis at two distinct points. If the discriminant is zero (${ \Delta = 0 }$), the equation has exactly one real root, which is a repeated root. In this case, the parabola touches the x-axis at exactly one point. However, when the discriminant is negative (${ \Delta < 0 }$), the equation has no real roots. This is because the square root of a negative number is not a real number. Instead, it leads to complex roots. Specifically, a negative discriminant implies that the parabola does not intersect the x-axis at any point. The roots, in this case, are a conjugate pair of complex numbers. Understanding this relationship between the discriminant and the nature of roots is crucial for solving and interpreting quadratic equations. It allows us to predict the type of solutions we will obtain even before applying the quadratic formula, saving time and effort. This knowledge is not just theoretical; it has practical applications in various fields where quadratic equations are used to model real-world phenomena.

Complex Solutions: A Deep Dive

When the discriminant (${ \Delta = b^2 - 4ac }$) is negative, we encounter complex solutions. These solutions involve the imaginary unit, denoted by ${ i }$, where ${ i = \sqrt{-1} }$. A complex number is generally expressed in the form ${ a + bi }$, where ${ a }$ is the real part and ${ b }$ is the imaginary part. The solutions to a quadratic equation with a negative discriminant are a pair of complex conjugates. This means that if one solution is ${ a + bi }$, the other solution is ${ a - bi }$. These complex solutions arise because we are taking the square root of a negative number, which is not possible within the realm of real numbers. The presence of complex solutions indicates that the parabola represented by the quadratic equation does not intersect the x-axis. Instead, it lies entirely above or below the x-axis. Complex solutions are not just mathematical abstractions; they have significant applications in various fields. In electrical engineering, they are used to analyze alternating current circuits. In quantum mechanics, they appear in the solutions to the Schrödinger equation. In control systems, they are used to analyze the stability of systems. Understanding complex solutions is essential for a complete understanding of quadratic equations and their applications. It broadens our mathematical toolkit and allows us to solve problems that would be unsolvable if we only considered real numbers. The ability to work with complex numbers is a valuable skill for anyone working in mathematics, science, or engineering. It opens up new possibilities for modeling and understanding the world around us.

Analyzing the Given Equations

Now, let's apply our understanding of the discriminant and complex solutions to the given equations. We will analyze each equation to determine whether it has no real solutions but two complex solutions. This involves calculating the discriminant for each equation and checking if it is negative. If the discriminant is negative, we can confidently conclude that the equation has two complex solutions. This process is not just about finding the solutions; it's about understanding the nature of the solutions and how they relate to the coefficients of the equation. It's about developing a deeper intuition for quadratic equations and their behavior. This analytical approach is crucial for problem-solving in mathematics and related fields. It allows us to break down complex problems into smaller, manageable steps and to apply the appropriate tools and techniques. In the following sections, we will systematically analyze each equation, showing the steps involved in calculating the discriminant and interpreting the results. This will provide a clear and concise illustration of how to determine whether an equation has complex solutions. This hands-on approach will solidify our understanding and equip us with the skills to tackle similar problems in the future. So, let's dive into the analysis and uncover the hidden nature of these equations.

Equation 1: ${ 3x^2 - 5x = -8 }$

To analyze the first equation, ${ 3x^2 - 5x = -8 }$, we first need to rewrite it in the standard quadratic form, which is ${ ax^2 + bx + c = 0 }$. Adding 8 to both sides, we get ${ 3x^2 - 5x + 8 = 0 }$. Now we can identify the coefficients: ${ a = 3 }$, ${ b = -5 }$, and ${ c = 8 }$. The discriminant, ${ \Delta }$, is calculated as ${ b^2 - 4ac }$. Substituting the values, we get ${ \Delta = (-5)^2 - 4(3)(8) = 25 - 96 = -71 }$. Since the discriminant is negative (${ -71 < 0 }$), this equation has no real solutions but two complex solutions. This means that the parabola represented by this equation does not intersect the x-axis. The solutions are a pair of complex conjugates, which can be found using the quadratic formula. This result highlights the importance of the discriminant in determining the nature of the roots. By simply calculating the discriminant, we can quickly determine whether an equation has real or complex solutions. This is a valuable tool in problem-solving, as it allows us to focus our efforts on the appropriate methods for finding the solutions. In this case, knowing that the solutions are complex, we would proceed to use the quadratic formula to find the complex conjugate pairs.

Equation 2: ${ 2x^2 = 6x - 5 }$

For the second equation, ${ 2x^2 = 6x - 5 }$, we again rewrite it in standard form: ${ 2x^2 - 6x + 5 = 0 }$. The coefficients are ${ a = 2 }$, ${ b = -6 }$, and ${ c = 5 }$. Calculating the discriminant, ${ \Delta = b^2 - 4ac = (-6)^2 - 4(2)(5) = 36 - 40 = -4 }$. As the discriminant is negative (${ -4 < 0 }$), this equation also has no real solutions but two complex solutions. Similar to the first equation, the parabola represented by this equation does not intersect the x-axis, and the solutions are a pair of complex conjugates. The negative discriminant confirms that the roots are complex, and the quadratic formula can be used to find these roots. This reinforces the concept that a negative discriminant is a clear indicator of complex solutions. By recognizing this pattern, we can quickly identify equations with complex solutions and apply the appropriate methods to solve them. This efficiency is crucial in mathematical problem-solving, especially in situations where time is a constraint. The ability to quickly assess the nature of the roots allows us to focus our efforts on the most effective solution strategies.

Equation 3: ${ 12x = 9x^2 + 4 }$

Let's analyze the third equation, ${ 12x = 9x^2 + 4 }$. Rewriting it in standard form, we get ${ 9x^2 - 12x + 4 = 0 }$. The coefficients are ${ a = 9 }$, ${ b = -12 }$, and ${ c = 4 }$. The discriminant is ${ \Delta = b^2 - 4ac = (-12)^2 - 4(9)(4) = 144 - 144 = 0 }$. Since the discriminant is zero (${ \Delta = 0 }$), this equation has one real solution (a repeated root). This means the parabola touches the x-axis at exactly one point. This case is different from the previous two, where the discriminant was negative. A zero discriminant indicates that the quadratic equation has a single, repeated real root. This is a special case where the parabola is tangent to the x-axis. The solution can be found using the quadratic formula, but since the discriminant is zero, the ${ \pm }$ term vanishes, resulting in a single root. This example highlights the importance of considering all possible cases when analyzing the discriminant. It's not just about whether the discriminant is positive or negative; the case where it's zero also provides valuable information about the nature of the roots.

Equation 4: ${ -x^2 - 10x = 34 }$

Finally, we analyze the fourth equation, ${ -x^2 - 10x = 34 }$. Rewriting it in standard form, we have ${ -x^2 - 10x - 34 = 0 }$. The coefficients are ${ a = -1 }$, ${ b = -10 }$, and ${ c = -34 }$. Calculating the discriminant, ${ \Delta = b^2 - 4ac = (-10)^2 - 4(-1)(-34) = 100 - 136 = -36 }$. The discriminant is negative (${ -36 < 0 }$), so this equation has no real solutions but two complex solutions. This result is consistent with the previous examples where a negative discriminant indicates complex roots. The parabola represented by this equation does not intersect the x-axis, and the solutions are a pair of complex conjugates. This final analysis reinforces the pattern we've observed throughout this exploration: a negative discriminant is a reliable indicator of complex solutions. By understanding this relationship, we can efficiently analyze quadratic equations and determine the nature of their roots.

Conclusion: Mastering Complex Solutions

In conclusion, we've explored the concept of quadratic equations with no real solutions but two complex solutions. We've learned that the discriminant, ${ \Delta = b^2 - 4ac }$, is the key to determining the nature of the roots. A negative discriminant indicates complex solutions, while a positive discriminant indicates real solutions, and a zero discriminant indicates a repeated real root. By analyzing the given equations, we've seen how to apply this knowledge in practice. We've identified equations with complex solutions by calculating their discriminants and finding them to be negative. This understanding is crucial for solving quadratic equations and for applying them in various fields. Complex solutions are not just abstract mathematical concepts; they have real-world applications in areas such as electrical engineering, quantum mechanics, and control systems. Mastering the concept of complex solutions broadens our mathematical toolkit and allows us to tackle a wider range of problems. It's a testament to the power and beauty of mathematics, which extends beyond the realm of real numbers to encompass the fascinating world of complex numbers. This exploration has hopefully demystified complex solutions and provided a solid foundation for further study in this area.