Solving The Quadratic Equation $3z^2 + 4z = -7$ A Step-by-Step Guide

Jul 10, 2025 by ADMIN 69 views

In this article, we will delve into the process of solving the quadratic equation $3z^2 + 4z = -7$ . Quadratic equations, which are polynomial equations of the second degree, are fundamental in mathematics and have wide-ranging applications in various fields, including physics, engineering, and economics. Understanding how to solve them is a crucial skill for anyone pursuing studies or work in these areas. We will explore the steps involved in finding the solutions (also known as roots) of this particular equation, providing a clear and comprehensive explanation along the way. Let's embark on this mathematical journey to find the values of z that satisfy the given equation.

Understanding Quadratic Equations

Before we dive into solving the specific equation, let's first establish a solid understanding of quadratic equations in general. A quadratic equation is a polynomial equation of the second degree, meaning that the highest power of the variable is 2. The standard form of a quadratic equation is expressed as:

$ax^2 + bx + c = 0$

where a, b, and c are constants, and x is the variable we aim to solve for. The coefficient a cannot be zero, as this would reduce the equation to a linear form. The solutions to a quadratic equation, also known as roots or zeros, are the values of x that make the equation true. These roots represent the points where the parabola defined by the quadratic equation intersects the x-axis on a graph. Quadratic equations can have two distinct real roots, one repeated real root, or two complex roots, depending on the discriminant ( $b^2 - 4ac$ ). Understanding these fundamental concepts is essential for tackling any quadratic equation, including the one we're about to solve.

Methods for Solving Quadratic Equations

There are several methods available for solving quadratic equations, each with its own strengths and applications. The most common methods include:

Factoring: This method involves expressing the quadratic expression as a product of two linear factors. If we can factor the quadratic equation into the form $(px + q)(rx + s) = 0$ , then the solutions are found by setting each factor equal to zero and solving for x. Factoring is a straightforward method when the quadratic expression can be easily factored, but it's not always applicable.
Completing the Square: This technique involves manipulating the quadratic equation to create a perfect square trinomial on one side. By adding and subtracting a specific constant, we can rewrite the equation in the form $(x + h)^2 = k$ , where h and k are constants. Taking the square root of both sides then allows us to solve for x. Completing the square is a powerful method that can be used to solve any quadratic equation.
Quadratic Formula: The quadratic formula is a universal method for solving quadratic equations. It provides a direct solution for x in terms of the coefficients a, b, and c. The formula is:

$x = rac{-b ime ext {pm} ext { ext {sqrt} {b^2 - 4ac}}}{2a}$

The quadratic formula is derived by completing the square on the general form of the quadratic equation. It's a reliable method, especially when factoring is difficult or impossible.
Graphical Method: Quadratic equations can also be solved graphically by plotting the corresponding quadratic function, $y = ax^2 + bx + c$ , and finding the x-intercepts (where the graph crosses the x-axis). These x-intercepts represent the real roots of the equation. The graphical method provides a visual representation of the solutions and can be useful for approximating roots, but it may not always provide exact solutions.

In our case, we will primarily use the quadratic formula to solve the given equation due to its general applicability and efficiency.

Applying the Quadratic Formula to $3z^2 + 4z = -7$

Now, let's apply the quadratic formula to solve the equation $3z^2 + 4z = -7$ . The first step is to rewrite the equation in the standard form, $ax^2 + bx + c = 0$ . To do this, we add 7 to both sides of the equation:

$3z^2 + 4z + 7 = 0$

Now we can identify the coefficients: a = 3, b = 4, and c = 7. Plugging these values into the quadratic formula, we get:

$z = rac{-b ime ext {pm} ext { ext {sqrt} {b^2 - 4ac}}}{2a} = rac{-4 ime ext {pm} ext { ext {sqrt} {4^2 - 4 imes 3 imes 7}}}{2 imes 3}$

Simplifying the expression under the square root:

$z = rac{-4 ime ext {pm} ext { ext {sqrt} {16 - 84}}}{6} = rac{-4 ime ext {pm} ext { ext {sqrt} {-68}}}{6}$

Since the value under the square root is negative, we have complex roots. We can rewrite the square root of -68 as:

$ ext { ext {sqrt} {-68}} = ext { ext {sqrt} {68}} imes ext { ext {sqrt} {-1}} = ext { ext {sqrt} {4 imes 17}} imes i = 2i ext { ext {sqrt} {17}}$

Substituting this back into the equation:

$z = rac{-4 ime ext {pm} 2i ext { ext {sqrt} {17}}}{6}$

Now, we can simplify by dividing both the numerator and the denominator by 2:

$z = rac{-2 ime ext {pm} i ext { ext {sqrt} {17}}}{3}$

Thus, the solutions to the quadratic equation are:

$z = rac{-2 + i ext { ext {sqrt} {17}}}{3}$ and $z = rac{-2 - i ext { ext {sqrt} {17}}}{3}$

Therefore, the correct answer is B. $rac{-2 ime ext {pm} i ext { ext {sqrt} {17}}}{3}$ . This detailed step-by-step solution demonstrates the application of the quadratic formula to solve a quadratic equation with complex roots. Understanding each step is crucial for mastering quadratic equations and their applications.

Alternative Methods and Considerations

While we effectively used the quadratic formula to solve the equation $3z^2 + 4z = -7$ , it's worth briefly exploring alternative methods and considerations that might be applicable in different scenarios. As mentioned earlier, factoring and completing the square are two other common techniques for solving quadratic equations. However, in this particular case, factoring is not straightforward due to the presence of complex roots, and attempting to factor would likely be more cumbersome than using the quadratic formula. Completing the square, while a valid method, involves more algebraic manipulation compared to the direct application of the quadratic formula.

Another consideration is the nature of the roots. In this case, we encountered complex roots, which arise when the discriminant ( $b^2 - 4ac$ ) is negative. Complex roots always occur in conjugate pairs, meaning that if $a + bi$ is a root, then $a - bi$ is also a root. Recognizing this property can be helpful in verifying solutions or identifying potential errors. Furthermore, understanding the discriminant can provide insights into the nature of the roots without actually solving the equation. A positive discriminant indicates two distinct real roots, a zero discriminant indicates one repeated real root, and a negative discriminant indicates two complex roots.

In practical applications, the choice of method for solving a quadratic equation often depends on the specific equation and the desired level of precision. The quadratic formula is generally the most reliable and versatile method, especially when dealing with equations that are difficult to factor or have complex roots. However, in simpler cases, factoring or completing the square might be more efficient. Ultimately, having a strong understanding of all these methods allows for a flexible and strategic approach to solving quadratic equations.

Real-World Applications of Quadratic Equations

Quadratic equations are not just abstract mathematical concepts; they have a wide range of real-world applications in various fields, making their study essential. One common application is in physics, where quadratic equations are used to describe projectile motion. The trajectory of a projectile, such as a ball thrown into the air, can be modeled using a quadratic equation, allowing us to calculate its height, range, and time of flight. Understanding quadratic equations is crucial for analyzing and predicting the behavior of objects in motion.

In engineering, quadratic equations are used in structural design to calculate stresses and strains in materials, ensuring the stability and safety of buildings and bridges. They also play a role in electrical engineering, where they are used to analyze circuits and determine optimal component values. The design of lenses and mirrors in optical systems also relies on quadratic equations to achieve the desired focusing and magnification properties.

Economics and finance also utilize quadratic equations. For example, they can be used to model cost and revenue functions, allowing businesses to determine the optimal production levels to maximize profit. Quadratic equations are also used in portfolio optimization to balance risk and return in investment strategies. The ability to solve quadratic equations is therefore valuable for making informed decisions in business and finance.

In addition to these specific examples, quadratic equations appear in many other areas, such as computer graphics, statistics, and even everyday problem-solving. Their versatility and widespread applicability make them a fundamental tool in mathematics and a crucial concept to master for anyone pursuing a career in science, technology, engineering, or mathematics (STEM) fields. By understanding the principles and techniques for solving quadratic equations, we gain the ability to model and analyze a wide range of real-world phenomena.

Conclusion

In conclusion, we have successfully solved the quadratic equation $3z^2 + 4z = -7$ using the quadratic formula. By rewriting the equation in standard form, identifying the coefficients, and applying the formula, we found the complex roots $z = rac{-2 + i ext { ext {sqrt} {17}}}{3}$ and $z = rac{-2 - i ext { ext {sqrt} {17}}}{3}$ . This process demonstrates the power and versatility of the quadratic formula in solving quadratic equations, even those with complex solutions. We also discussed alternative methods, such as factoring and completing the square, and highlighted their applicability in different scenarios.

Furthermore, we emphasized the real-world applications of quadratic equations in fields such as physics, engineering, and economics, underscoring their importance in various disciplines. From modeling projectile motion to designing structures and optimizing financial strategies, quadratic equations provide a valuable framework for analyzing and solving a wide range of problems.

By mastering the techniques for solving quadratic equations and understanding their applications, students and professionals alike can enhance their problem-solving skills and gain a deeper appreciation for the role of mathematics in the world around us. This article serves as a comprehensive guide to solving quadratic equations, providing a solid foundation for further exploration of mathematical concepts and their practical applications. The ability to confidently tackle quadratic equations is a valuable asset in any STEM field and beyond, empowering individuals to approach complex problems with clarity and precision.