Solving $-2x^2 + 3x - 9 = 0$ Complex Solutions Explained

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In this comprehensive guide, we will delve into the intricate process of solving the quadratic equation $-2x^2 + 3x - 9 = 0$. Quadratic equations, characterized by their second-degree polynomial nature, frequently arise in various mathematical and scientific contexts. This particular equation presents an interesting case due to its potential for complex solutions, which we will explore in detail. To effectively tackle this challenge, we will employ the renowned quadratic formula, a cornerstone of algebraic problem-solving.

The quadratic formula serves as our primary tool for determining the roots (or solutions) of any quadratic equation expressed in the standard form of $ax^2 + bx + c = 0$. The formula is elegantly expressed as:

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

where:

• $x$ represents the roots of the equation.
• $a$, $b$, and $c$ are the coefficients of the quadratic equation. In our specific case, $a = -2$, $b = 3$, and $c = -9$.

Before we plunge into the calculations, let's first discuss the significance of the discriminant, which is the expression under the square root in the quadratic formula, $b^2 - 4ac$. The discriminant provides invaluable information about the nature of the roots. If the discriminant is positive, the equation has two distinct real roots. If it is zero, the equation has exactly one real root (a repeated root). However, if the discriminant is negative, as we will soon discover in our case, the equation possesses two complex conjugate roots. Complex roots involve the imaginary unit, denoted by $i$, where $i^2 = -1$. These roots are expressed in the form $p + qi$, where $p$ and $q$ are real numbers and $i$ is the imaginary unit.

Now, let's proceed with the calculations. We will substitute the values of $a$, $b$, and $c$ into the quadratic formula and meticulously simplify the expression to reveal the solutions.

To solve the quadratic equation $-2x^2 + 3x - 9 = 0$, we begin by identifying the coefficients: $a = -2$, $b = 3$, and $c = -9$. We then substitute these values into the quadratic formula:

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

Plugging in the values, we get:

$$x = \frac{-3 \pm \sqrt{3^2 - 4(-2)(-9)}}{2(-2)}$$

Let's simplify the expression step by step. First, we calculate the discriminant:

$$3^2 - 4(-2)(-9) = 9 - 72 = -63$$

As anticipated, the discriminant is negative (-63), confirming that the equation has complex roots. Now we can substitute this value back into the quadratic formula:

$$x = \frac{-3 \pm \sqrt{-63}}{-4}$$

Since we have a negative value under the square root, we introduce the imaginary unit $i$. Recall that $\sqrt{-1} = i$. We can rewrite $\sqrt{-63}$ as $\sqrt{63}i$. Furthermore, we can simplify $\sqrt{63}$ by factoring out the largest perfect square, which is 9:

$$\sqrt{63} = \sqrt{9 \cdot 7} = \sqrt{9} \cdot \sqrt{7} = 3\sqrt{7}$$

Therefore, $\sqrt{-63} = 3i\sqrt{7}$. Substituting this back into the equation, we have:

$$x = \frac{-3 \pm 3i\sqrt{7}}{-4}$$

To simplify further, we can divide both the numerator and the denominator by -1:

$$x = \frac{3 \mp 3i\sqrt{7}}{4}$$

This expression gives us two complex solutions:

$$x_1 = \frac{3 + 3i\sqrt{7}}{4}$$

$$x_2 = \frac{3 - 3i\sqrt{7}}{4}$$

These complex solutions are conjugates of each other, meaning they have the same real part but opposite imaginary parts.

Having obtained the complex solutions $x_1 = \frac{3 + 3i\sqrt{7}}{4}$ and $x_2 = \frac{3 - 3i\sqrt{7}}{4}$, it is crucial to understand their implications and significance within the context of quadratic equations. These solutions highlight a fundamental property: when a quadratic equation with real coefficients has complex roots, these roots always occur in conjugate pairs.

Complex conjugate roots emerge when the discriminant of the quadratic equation ($b^2 - 4ac$) is negative. In our case, the discriminant was calculated to be -63, a clear indicator of complex roots. The conjugate nature of the roots stems from the $\pm$ sign in the quadratic formula. This sign dictates that the imaginary part of one root will be the opposite of the imaginary part of the other root, while the real parts remain identical.

The complex roots indicate that the parabola represented by the quadratic equation $-2x^2 + 3x - 9 = 0$ does not intersect the x-axis. In the realm of real numbers, the roots of a quadratic equation correspond to the x-intercepts of its parabolic graph. However, when the roots are complex, the parabola hovers either entirely above or entirely below the x-axis, signifying the absence of real roots.

Let's delve deeper into the structure of our complex solutions. Each solution comprises a real part and an imaginary part. For $x_1 = \frac{3 + 3i\sqrt{7}}{4}$, the real part is $\frac{3}{4}$, and the imaginary part is $\frac{3\sqrt{7}}{4}$. Similarly, for $x_2 = \frac{3 - 3i\sqrt{7}}{4}$, the real part remains $\frac{3}{4}$, but the imaginary part is $-\frac{3\sqrt{7}}{4}$. The difference in the sign of the imaginary part is what characterizes them as complex conjugates.

In various fields, complex numbers play a pivotal role. They are extensively used in electrical engineering, quantum mechanics, and signal processing. The ability to solve quadratic equations with complex solutions empowers us to model and analyze phenomena in these domains.

To ensure the accuracy of our solutions, we can substitute them back into the original equation, $-2x^2 + 3x - 9 = 0$. If our solutions are correct, the equation should hold true (i.e., the left-hand side should equal the right-hand side, which is 0).

Let's begin by substituting $x_1 = \frac{3 + 3i\sqrt{7}}{4}$ into the equation:

$$-2\left(\frac{3 + 3i\sqrt{7}}{4}\right)^2 + 3\left(\frac{3 + 3i\sqrt{7}}{4}\right) - 9$$

We must carefully expand and simplify this expression. First, let's square the complex number:

$$\left(\frac{3 + 3i\sqrt{7}}{4}\right)^2 = \frac{(3 + 3i\sqrt{7})^2}{16} = \frac{9 + 18i\sqrt{7} - 63}{16} = \frac{-54 + 18i\sqrt{7}}{16}$$

Now, substitute this back into the equation:

$$-2\left(\frac{-54 + 18i\sqrt{7}}{16}\right) + 3\left(\frac{3 + 3i\sqrt{7}}{4}\right) - 9$$

Simplify each term:

$$\frac{108 - 36i\sqrt{7}}{16} + \frac{9 + 9i\sqrt{7}}{4} - 9$$

To combine these terms, we need a common denominator, which is 16. Convert the fractions accordingly:

$$\frac{108 - 36i\sqrt{7}}{16} + \frac{36 + 36i\sqrt{7}}{16} - \frac{144}{16}$$

Now, combine the numerators:

$$\frac{108 - 36i\sqrt{7} + 36 + 36i\sqrt{7} - 144}{16} = \frac{0}{16} = 0$$

The equation holds true for $x_1$. A similar process can be followed to verify the solution $x_2 = \frac{3 - 3i\sqrt{7}}{4}$. The verification process solidifies our confidence in the correctness of the solutions.

While the quadratic formula is a universally applicable method for solving quadratic equations, other techniques exist that can sometimes provide a more efficient or insightful approach. Two prominent methods are factoring and completing the square. Although these methods may not always be straightforward for every quadratic equation, they offer valuable alternative perspectives and can be particularly effective in specific scenarios.

Factoring

Factoring involves expressing the quadratic equation as a product of two linear factors. This method relies on the principle that if the product of two factors is zero, then at least one of the factors must be zero. Factoring is most effective when the quadratic equation has integer roots or roots that can be easily determined. However, for equations with complex roots or irrational roots, factoring can be challenging.

To illustrate, consider a simpler quadratic equation, such as $x^2 - 5x + 6 = 0$. We seek two numbers that multiply to 6 and add up to -5. These numbers are -2 and -3. Therefore, we can factor the equation as:

$$(x - 2)(x - 3) = 0$$

Setting each factor equal to zero, we find the roots $x = 2$ and $x = 3$.

However, our original equation, $-2x^2 + 3x - 9 = 0$, does not lend itself easily to factoring due to the presence of complex roots. The discriminant, as we previously calculated, is negative, indicating that the roots are not real numbers. In such cases, the quadratic formula remains the more reliable method.

Completing the Square

Completing the square is another powerful technique for solving quadratic equations. This method involves transforming the quadratic equation into a perfect square trinomial, which can then be easily solved by taking the square root. Completing the square is particularly useful when deriving the quadratic formula itself. It also provides a deeper understanding of the structure of quadratic equations and the relationship between their coefficients and roots.

The general procedure for completing the square involves several steps. First, we ensure that the coefficient of the $x^2$ term is 1. If it is not, we divide the entire equation by that coefficient. Then, we take half of the coefficient of the $x$ term, square it, and add it to both sides of the equation. This process creates a perfect square trinomial on one side of the equation, which can be factored as a squared binomial. Finally, we take the square root of both sides and solve for $x$.

Let's apply completing the square to the equation $x^2 + 4x - 5 = 0$. First, we take half of the coefficient of the $x$ term (which is 4), giving us 2. Squaring this value yields 4. Adding 4 to both sides of the equation, we get:

$$x^2 + 4x + 4 = 9$$

The left side is now a perfect square trinomial, which can be factored as $(x + 2)^2$:

$$(x + 2)^2 = 9$$

Taking the square root of both sides, we obtain:

$$x + 2 = \pm 3$$

Solving for $x$, we find the roots $x = 1$ and $x = -5$.

Applying completing the square to our original equation, $-2x^2 + 3x - 9 = 0$, would involve similar steps but would ultimately lead to the same complex solutions we obtained using the quadratic formula. While completing the square is a valuable technique, the quadratic formula often provides a more direct route to the solutions, especially when dealing with complex roots.

In addition to the algebraic methods we've explored, visualizing the quadratic equation graphically offers valuable insights into the nature of its solutions. The graph of a quadratic equation in the form $ax^2 + bx + c = 0$ is a parabola. The roots of the equation correspond to the x-intercepts of the parabola, which are the points where the parabola intersects the x-axis.

When the roots are real and distinct, the parabola intersects the x-axis at two distinct points. When the roots are real and equal (a repeated root), the parabola touches the x-axis at exactly one point, the vertex of the parabola. However, when the roots are complex, as in our case with the equation $-2x^2 + 3x - 9 = 0$, the parabola does not intersect the x-axis at all.

The parabola for the equation $-2x^2 + 3x - 9 = 0$ opens downwards because the coefficient of the $x^2$ term ($a$) is negative (-2). The vertex of the parabola represents the maximum point on the graph. Since the roots are complex, the entire parabola lies below the x-axis. This graphical representation visually confirms that there are no real solutions to the equation, consistent with our algebraic findings.

The complex roots can be visualized in the complex plane, which is a two-dimensional plane where the horizontal axis represents the real part of a complex number and the vertical axis represents the imaginary part. The complex roots $x_1 = \frac{3 + 3i\sqrt{7}}{4}$ and $x_2 = \frac{3 - 3i\sqrt{7}}{4}$ can be plotted as points in the complex plane. They will be located symmetrically with respect to the real axis, reflecting their conjugate relationship. This graphical representation further reinforces the understanding of complex solutions and their properties.

In summary, the graphical interpretation of quadratic equations provides a visual complement to the algebraic solutions. It allows us to connect the roots of the equation with the behavior of the parabola and to understand the implications of complex roots in terms of the graph's position relative to the x-axis. This multifaceted approach enhances our comprehension of quadratic equations and their solutions.

In conclusion, we have successfully solved the quadratic equation $-2x^2 + 3x - 9 = 0$ using the quadratic formula. We determined that the equation has two complex conjugate solutions: $x_1 = \frac{3 + 3i\sqrt{7}}{4}$ and $x_2 = \frac{3 - 3i\sqrt{7}}{4}$. These solutions underscore the importance of the discriminant in determining the nature of the roots. A negative discriminant signifies complex roots, which always occur in conjugate pairs for quadratic equations with real coefficients.

We further explored alternative methods for solving quadratic equations, including factoring and completing the square. While these methods can be effective in certain cases, the quadratic formula remains a robust and universally applicable tool, especially when dealing with complex roots. We also examined the graphical interpretation of the solutions, noting that complex roots correspond to parabolas that do not intersect the x-axis.

The solutions obtained in this analysis have profound implications in various scientific and engineering domains. Complex numbers, the foundation of these solutions, are instrumental in analyzing oscillatory phenomena, alternating current circuits, and quantum mechanical systems. A thorough understanding of complex numbers and their applications is invaluable for professionals in these fields.

By mastering the techniques for solving quadratic equations, we equip ourselves with a fundamental tool for mathematical problem-solving. The quadratic formula, in particular, stands as a testament to the power of algebraic methods in unraveling the complexities of equations. This knowledge empowers us to tackle a wide range of mathematical challenges and to apply these concepts in diverse real-world applications.

In summary, the exploration of the quadratic equation $-2x^2 + 3x - 9 = 0$ has provided a comprehensive insight into the nature of quadratic equations, complex solutions, and the diverse methods available for solving them. This understanding serves as a cornerstone for further mathematical endeavors and applications in various scientific and engineering disciplines.