Solving 3x² - 11x - 9 = 4 Real And Imaginary Solutions
Introduction
In this article, we will delve into the process of finding all solutions, both real and imaginary, for the quadratic equation 3x² - 11x - 9 = 4. We will employ a method of choice to solve this equation and, if necessary, round the solutions to three decimal places. Understanding how to solve quadratic equations is a fundamental skill in algebra, with applications spanning various fields, including physics, engineering, and computer science. Quadratic equations, characterized by the presence of a term with the variable raised to the power of two, often arise in modeling real-world phenomena, such as projectile motion, electrical circuits, and financial investments.
Before diving into the solution process, let's first understand the general form of a quadratic equation and the different methods available for solving them. A quadratic equation is typically expressed in the standard form as ax² + bx + c = 0, where a, b, and c are constants, and x represents the variable we aim to solve for. The solutions to a quadratic equation are also known as its roots or zeros, representing the values of x that satisfy the equation. There are several methods to find these solutions, each with its own advantages and applicability depending on the specific equation. Factoring, completing the square, and the quadratic formula are the most commonly used methods. Factoring is a straightforward technique applicable when the quadratic expression can be easily factored into two linear expressions. Completing the square involves transforming the equation into a perfect square trinomial, making it easier to solve. The quadratic formula is a universal method that provides solutions for any quadratic equation, regardless of its factorability. In this article, we will use the quadratic formula to solve the given equation, as it is a robust method that guarantees a solution, even when dealing with complex or irrational roots.
Transforming the Equation to Standard Form
To begin, it's essential to rearrange the given equation, 3x² - 11x - 9 = 4, into the standard quadratic form, which is ax² + bx + c = 0. This form allows us to easily identify the coefficients a, b, and c, which are crucial for applying the quadratic formula or other solution methods. By subtracting 4 from both sides of the equation, we can achieve the standard form. This step ensures that all terms are on one side of the equation, leaving zero on the other side, which is the required format for solving quadratic equations. The process of rearranging equations is a fundamental algebraic skill, and it is particularly important when dealing with quadratic equations, as the standard form provides a consistent framework for applying various solution techniques. The ability to manipulate equations into the desired form is essential for simplifying complex mathematical problems and making them more amenable to solution. In the context of quadratic equations, the standard form not only facilitates the application of solution methods but also provides insights into the nature of the roots, such as whether they are real or complex, and their multiplicity.
So, we subtract 4 from both sides:
3x² - 11x - 9 - 4 = 4 - 4
This simplifies to:
3x² - 11x - 13 = 0
Now, we have the equation in the standard form, where:
- a = 3
- b = -11
- c = -13
Applying the Quadratic Formula
Now that we have the quadratic equation in standard form, we can apply the quadratic formula to find the solutions for x. The quadratic formula is a powerful tool that provides a general solution for any quadratic equation of the form ax² + bx + c = 0. It is derived by completing the square on the standard form equation and is given by:
x = (-b ± √(b² - 4ac)) / (2a)
This formula is a cornerstone of algebra, and its versatility makes it an indispensable tool for solving quadratic equations. The quadratic formula guarantees a solution, regardless of whether the roots are real or complex, rational or irrational. It is particularly useful when factoring is difficult or impossible, and it provides a systematic approach to finding solutions. The formula involves substituting the coefficients a, b, and c from the standard form equation into the formula and simplifying the expression. The ± symbol indicates that there are two possible solutions, one obtained by adding the square root term and the other by subtracting it. The expression inside the square root, b² - 4ac, is known as the discriminant, and it provides valuable information about the nature of the roots. A positive discriminant indicates two distinct real roots, a zero discriminant indicates one real root (a repeated root), and a negative discriminant indicates two complex roots.
Let's substitute the values of a, b, and c from our equation (3x² - 11x - 13 = 0) into the quadratic formula:
x = (-(-11) ± √((-11)² - 4 * 3 * (-13))) / (2 * 3)
Simplifying the expression:
x = (11 ± √(121 + 156)) / 6
x = (11 ± √277) / 6
Calculating the Solutions
Now, we need to calculate the two possible solutions for x by considering both the addition and subtraction of the square root term. The square root of 277 is not a perfect square, so we will obtain irrational solutions. To find the approximate values, we will use a calculator and round the results to three decimal places as requested. This step is crucial for obtaining numerical solutions that can be used in practical applications. The process of approximating irrational numbers is an essential skill in mathematics, and it allows us to work with solutions that may not have exact decimal representations. Rounding to a specific number of decimal places ensures that the solutions are presented with a consistent level of precision, making them easier to interpret and compare. In the context of quadratic equations, approximate solutions are often sufficient for practical purposes, such as in engineering or physics, where measurements and calculations are often subject to some degree of uncertainty.
First, let's calculate the solution with addition:
x₁ = (11 + √277) / 6
Using a calculator, we find:
x₁ ≈ (11 + 16.643) / 6
x₁ ≈ 27.643 / 6
x₁ ≈ 4.607
Next, let's calculate the solution with subtraction:
x₂ = (11 - √277) / 6
Using a calculator, we find:
x₂ ≈ (11 - 16.643) / 6
x₂ ≈ -5.643 / 6
x₂ ≈ -0.941
Final Answer
Therefore, the solutions to the quadratic equation 3x² - 11x - 9 = 4, rounded to three decimal places, are:
x₁ ≈ 4.607 and x₂ ≈ -0.941
These two values of x are the points where the quadratic function intersects the x-axis, representing the roots or zeros of the equation. The solutions are real and distinct, which is consistent with the positive discriminant we calculated earlier. The ability to find these solutions is essential for understanding the behavior of the quadratic function and its applications in various fields. The solutions can be used to determine the vertex of the parabola, the axis of symmetry, and the intervals where the function is positive or negative. In practical applications, these solutions may represent physical quantities, such as time, distance, or concentration, depending on the context of the problem.
Conclusion
In summary, we have successfully found the solutions to the quadratic equation 3x² - 11x - 9 = 4 by first transforming it into standard form and then applying the quadratic formula. This method allowed us to determine both real solutions, which we rounded to three decimal places as requested. The quadratic formula is a powerful and versatile tool for solving quadratic equations, providing a systematic approach that guarantees a solution, regardless of the nature of the roots. Understanding how to apply the quadratic formula is a fundamental skill in algebra, with applications in various fields of science, engineering, and mathematics. The ability to solve quadratic equations is essential for modeling and analyzing real-world phenomena, and it forms the basis for more advanced mathematical concepts. The process we followed in this article, from transforming the equation to standard form to applying the quadratic formula and calculating the solutions, provides a clear and concise framework for solving any quadratic equation. By mastering this process, you will be well-equipped to tackle a wide range of mathematical problems and applications involving quadratic equations.
