Solving Quadratic Equations Finding The Roots Of -10x^2 + 12x - 9 = 0
In the fascinating realm of mathematics, quadratic equations hold a special place. These equations, characterized by their second-degree polynomial form, appear in various scientific and engineering applications. Understanding how to solve them is a fundamental skill for any aspiring mathematician or scientist. This article aims to delve into the intricacies of solving the quadratic equation -10x^2 + 12x - 9 = 0, providing a step-by-step guide and explaining the underlying concepts. We'll explore the quadratic formula, a powerful tool for finding the roots of any quadratic equation, and carefully analyze the discriminant to determine the nature of the roots. This comprehensive guide will equip you with the knowledge and skills needed to confidently tackle similar problems. This exploration of quadratic equations begins with understanding its form, the ax^2 + bx + c = 0 template, that helps us identify the coefficients a, b, and c, which are crucial for applying the quadratic formula. This article will not only provide a solution but also deepen your understanding of the mathematical principles at play. Quadratic equations, with their characteristic parabolic curves when graphed, are more than just abstract mathematical constructs; they are models for projectile motion, optimization problems, and numerous other real-world phenomena. Therefore, mastering the techniques to solve them opens doors to a broader understanding of the world around us.
H2: Understanding Quadratic Equations
A quadratic equation is a polynomial equation of the second degree. Its general form is expressed as ax^2 + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' represents the unknown variable. The coefficients 'a', 'b', and 'c' play a crucial role in determining the nature and values of the roots of the equation. The term 'a' cannot be zero; otherwise, the equation would reduce to a linear equation. In our specific case, the equation -10x^2 + 12x - 9 = 0 fits this form perfectly, allowing us to apply the standard methods for solving quadratic equations. Recognizing the form of a quadratic equation is the first step in solving it. This form allows us to identify the coefficients 'a', 'b', and 'c' which are the keys to unlocking the solutions. Understanding the role of each coefficient is crucial. The coefficient 'a' determines the parabola's direction (whether it opens upwards or downwards) and its width. The coefficient 'b' affects the parabola's position along the x-axis, and the constant 'c' represents the y-intercept of the parabola. When we delve into the methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula, these coefficients are the foundation upon which we build our solution strategy. The quadratic formula, perhaps the most versatile method, directly incorporates these coefficients into its formula, providing a straightforward way to find the roots. The discriminant, a part of the quadratic formula, further uses these coefficients to tell us about the nature of the roots ā whether they are real and distinct, real and equal, or complex conjugates. Thus, the general form ax^2 + bx + c = 0 is not just a notational convenience; it is the bedrock of our understanding and solution process for quadratic equations. Understanding these basic concepts is crucial before diving into solving the equation -10x^2 + 12x - 9 = 0. We need to clearly identify the values of a, b, and c in this particular equation to apply the appropriate solution method.
H2: The Quadratic Formula The Key to Finding Roots
The quadratic formula is a powerful tool for finding the roots of any quadratic equation. It is derived from the method of completing the square and provides a direct solution for 'x' in terms of the coefficients 'a', 'b', and 'c'. The formula is given by:
x = (-b Ā± ā(b^2 - 4ac)) / (2a)
This formula is a cornerstone of algebra and is widely used in various fields. The expression inside the square root, b^2 - 4ac, is known as the discriminant. The discriminant plays a vital role in determining the nature of the roots, and we'll discuss it in detail later. In essence, the quadratic formula is a universal key that unlocks the solutions to any quadratic equation, regardless of its complexity. It elegantly captures the relationship between the coefficients of the equation and its roots, providing a systematic way to find the values of 'x' that satisfy the equation. What makes the quadratic formula so powerful is its ability to handle any quadratic equation, even those that are difficult or impossible to solve by factoring. The derivation of the quadratic formula, through the process of completing the square on the general form ax^2 + bx + c = 0, showcases the beauty and elegance of mathematical reasoning. Each step in the derivation reveals how the coefficients are manipulated to isolate 'x' and express it in terms of 'a', 'b', and 'c'. Beyond its practical utility, the quadratic formula is a testament to the power of mathematical abstraction and generalization. It exemplifies how a single formula can encapsulate a fundamental concept and provide a universal solution to a class of problems. Understanding the quadratic formula is not just about memorizing a formula; it's about appreciating the underlying mathematical principles and recognizing its applicability in diverse contexts. From physics to engineering, from economics to computer science, the quadratic formula finds its place in solving a multitude of real-world problems.
H2: Applying the Quadratic Formula to -10x^2 + 12x - 9 = 0
To solve the equation -10x^2 + 12x - 9 = 0 using the quadratic formula, we first identify the coefficients: a = -10, b = 12, and c = -9. Then, we substitute these values into the formula:
x = (-12 Ā± ā(12^2 - 4(-10)(-9))) / (2(-10))
x = (-12 Ā± ā(144 - 360)) / (-20)
x = (-12 Ā± ā(-216)) / (-20)
This substitution is the core of the solution process, transforming the abstract formula into a concrete calculation. We carefully replace each coefficient in the formula with its corresponding value from the equation, paying close attention to signs and order of operations. The next step involves simplifying the expression under the square root, which is the discriminant. In this case, the discriminant is negative, indicating that the roots will be complex numbers. The simplification process requires meticulous arithmetic, ensuring that each operation is performed correctly. We square 'b', multiply 4ac, and then subtract the latter from the former. The result, in this case, is a negative number, which immediately tells us that we will be dealing with complex roots. The denominator, 2a, is also calculated, providing the final piece of the puzzle for applying the quadratic formula. Now, with the discriminant simplified and the denominator calculated, we are ready to express the roots in their final form. The Ā± sign in the formula indicates that there are two roots, one obtained by adding the square root of the discriminant and the other by subtracting it. This reflects the fundamental nature of quadratic equations, which generally have two solutions. The quadratic formula, when applied correctly, guides us through this entire process, ensuring that we arrive at the correct roots, even when they are complex numbers. The process of applying the quadratic formula is not just a mechanical exercise; it is a journey through the heart of a quadratic equation, revealing its hidden solutions and showcasing the power of mathematical tools.
H2: Simplifying the Roots and Finding the Solution
Now, let's simplify the expression further. We have:
x = (-12 Ā± ā(-216)) / (-20)
Since ā(-216) = ā(216 * -1) = ā(36 * 6 * -1) = 6iā6, we can rewrite the equation as:
x = (-12 Ā± 6iā6) / (-20)
Dividing both the numerator and the denominator by -4, we get:
x = (3 Ā± (-3/2)iā6) / 5
x = 3/5 Ā± (3iā6) / 10
Therefore, the roots of the quadratic equation -10x^2 + 12x - 9 = 0 are:
x = 3/5 + (3iā6) / 10 and x = 3/5 - (3iā6) / 10
Thus, the correct answer is D. x = 3/5 Ā± (3iā6) / 10. Simplifying complex numbers requires a solid understanding of imaginary units and their properties. The imaginary unit, denoted by 'i', is defined as the square root of -1. This seemingly simple concept unlocks a whole new dimension in mathematics, allowing us to deal with the square roots of negative numbers. When we encounter a negative number under a square root, as in our case with ā(-216), we can express it as the product of -1 and the positive number, and then extract the square root of -1 as 'i'. This allows us to transform ā(-216) into 6iā6. The ability to manipulate and simplify complex numbers is crucial in many areas of mathematics and physics. From electrical engineering to quantum mechanics, complex numbers provide a powerful tool for modeling and understanding a wide range of phenomena. In our context, simplifying the roots involves not only extracting 'i' but also finding the largest perfect square factor within the remaining radical. This allows us to express the radical in its simplest form, making the roots more manageable and easier to interpret. The final step in simplifying the roots involves dividing both the real and imaginary parts of the numerator by the denominator. This ensures that the roots are expressed in their standard form, making it easier to compare them with answer choices or to use them in further calculations. The process of simplifying complex roots is a testament to the elegance and precision of mathematical notation. It allows us to transform seemingly complicated expressions into simpler, more manageable forms, revealing the underlying beauty and structure of the solution.
H2: The Discriminant Unveiling the Nature of Roots
The discriminant (b^2 - 4ac) provides valuable information about the nature of the roots of a quadratic equation. The discriminant helps determine whether the roots are real and distinct, real and equal, or complex conjugates:
- If b^2 - 4ac > 0, the equation has two distinct real roots.
- If b^2 - 4ac = 0, the equation has one real root (a repeated root).
- If b^2 - 4ac < 0, the equation has two complex conjugate roots.
In our case, the discriminant is 12^2 - 4(-10)(-9) = 144 - 360 = -216, which is less than 0. This confirms that the roots are complex conjugates, as we found in our solution. The discriminant is like a window into the soul of a quadratic equation, revealing the hidden characteristics of its roots. It is a single number, yet it holds a wealth of information about the solutions to the equation. By simply calculating the discriminant, we can determine whether the roots will be real or complex, distinct or repeated, without even solving the equation itself. This is a powerful tool for understanding the behavior of quadratic equations and for predicting the nature of their solutions. The relationship between the discriminant and the roots can be visualized graphically. A positive discriminant corresponds to a parabola that intersects the x-axis at two distinct points, representing the two real roots. A zero discriminant corresponds to a parabola that touches the x-axis at exactly one point, representing the repeated real root. A negative discriminant corresponds to a parabola that does not intersect the x-axis at all, indicating that the roots are complex conjugates. Understanding the discriminant not only helps us solve quadratic equations but also provides insights into the geometric interpretation of the solutions. It connects the algebraic properties of the equation to the visual representation of its graph, enriching our understanding of quadratic relationships. In practical applications, the discriminant can be used to quickly assess the feasibility of solutions. For example, in an engineering problem, if the discriminant is negative, it may indicate that the problem has no real-world solution and that the parameters need to be adjusted. Thus, the discriminant is a versatile tool that not only helps us solve equations but also provides valuable insights into the nature of the problems we are trying to solve.
H2: Conclusion Mastering Quadratic Equations
In this article, we have thoroughly explored the process of solving the quadratic equation -10x^2 + 12x - 9 = 0. We started by understanding the basic form of a quadratic equation, then delved into the quadratic formula, applied it to our specific equation, simplified the roots, and finally, used the discriminant to confirm the nature of the roots. This step-by-step approach demonstrates the power and elegance of mathematical problem-solving. Mastering quadratic equations is not just about finding the correct answers; it's about developing a deeper understanding of mathematical concepts and honing problem-solving skills. Quadratic equations are fundamental building blocks in mathematics, and their applications extend far beyond the classroom. From physics to engineering, from economics to computer science, quadratic equations play a crucial role in modeling and solving real-world problems. The skills we have developed in solving this equation ā understanding the quadratic formula, simplifying complex numbers, and interpreting the discriminant ā are transferable to a wide range of mathematical and scientific disciplines. This comprehensive guide is designed not only to help you solve specific equations but also to empower you with the knowledge and confidence to tackle any quadratic equation that comes your way. By understanding the underlying principles and mastering the techniques, you can unlock the secrets of quadratic equations and apply them to solve a multitude of problems. The journey of mastering quadratic equations is a journey of mathematical discovery and growth, opening doors to new possibilities and enriching our understanding of the world around us.
The final answer is D. x = 3/5 Ā± (3iā6) / 10.
