Solving 3x^2 - 5x - 7 = 0 A Step-by-Step Guide
Introduction
In mathematics, quadratic equations play a pivotal role in various fields, including algebra, calculus, and physics. A quadratic equation is a polynomial equation of the second degree, generally expressed in the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. Solving quadratic equations is a fundamental skill in mathematics, with numerous applications in real-world problems. In this article, we will delve into the process of solving the specific quadratic equation 3x² - 5x - 7 = 0, focusing on finding solutions to three significant figures. This involves understanding the quadratic formula and its application, as well as the importance of accurate rounding in mathematical calculations. This comprehensive guide aims to provide a clear, step-by-step approach to solving quadratic equations, ensuring a solid grasp of the underlying concepts and techniques. Whether you are a student tackling algebra problems or someone looking to refresh your mathematical skills, this article offers a valuable resource for mastering quadratic equation solutions.
Understanding Quadratic Equations
Before we dive into solving the specific equation, it's crucial to understand the fundamentals of quadratic equations. A quadratic equation is defined by its highest power of the variable being 2. The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are coefficients, and x is the variable. The coefficient a determines the shape of the parabola when the equation is graphed, b influences the position and direction, and c shifts the parabola vertically. Understanding these coefficients is essential for interpreting and solving quadratic equations effectively. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. Factoring involves breaking down the quadratic expression into simpler factors, which can be straightforward for some equations but challenging for others. Completing the square is a method that transforms the equation into a perfect square trinomial, making it easier to solve. However, the most versatile and widely used method is the quadratic formula, which can solve any quadratic equation, regardless of its complexity. This formula provides a direct way to find the roots of the equation, making it an indispensable tool in algebra. In the following sections, we will focus on applying the quadratic formula to solve the given equation, ensuring accurate results to three significant figures. By mastering this method, you will be well-equipped to tackle a wide range of quadratic equation problems.
The Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations of the form ax² + bx + c = 0. The formula is given by:
x = (-b ± √(b² - 4ac)) / (2a)
This formula provides the solutions (or roots) for x in terms of the coefficients a, b, and c. The expression inside the square root, b² - 4ac, is known as the discriminant. The discriminant plays a crucial role in determining 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). If the discriminant is negative, the equation has two complex roots. Understanding the discriminant helps in predicting the type of solutions to expect before actually applying the formula. The ± symbol in the quadratic formula indicates that there are generally two solutions, one obtained by adding the square root and the other by subtracting it. These solutions represent the points where the parabola intersects the x-axis, which are the roots of the equation. The quadratic formula is derived by completing the square on the general form of the quadratic equation, making it a reliable and universally applicable method. In the next sections, we will apply this formula to our specific equation, 3x² - 5x - 7 = 0, to find its solutions. Mastering the quadratic formula is essential for anyone studying algebra, as it provides a direct and efficient way to solve any quadratic equation.
Applying the Quadratic Formula to 3x^2 - 5x - 7 = 0
To solve the quadratic equation 3x² - 5x - 7 = 0 using the quadratic formula, we first need to identify the coefficients a, b, and c. In this equation:
- a = 3
- b = -5
- c = -7
Now, we substitute these values into the quadratic formula:
x = (-(-5) ± √((-5)² - 4(3)(-7))) / (2(3))
This simplifies to:
x = (5 ± √(25 + 84)) / 6
Further simplification gives us:
x = (5 ± √109) / 6
At this stage, we have two possible solutions for x, one with the addition of the square root and one with the subtraction. These are:
- x₁ = (5 + √109) / 6
- x₂ = (5 - √109) / 6
These represent the exact solutions to the equation. In the following section, we will calculate these values and round them to three significant figures, as required. This step is crucial for providing a practical and accurate answer, especially in applications where precision is important. By carefully substituting the coefficients and simplifying the expression, we have successfully set up the equation for the final calculation. This process highlights the power and efficiency of the quadratic formula in solving quadratic equations. The next step involves using a calculator to find the numerical values of the roots to the desired level of accuracy.
Calculating the Solutions
Now that we have the solutions in the form x₁ = (5 + √109) / 6 and x₂ = (5 - √109) / 6, we need to calculate these values and round them to three significant figures. Using a calculator, we find:
- x₁ = (5 + √109) / 6 ≈ (5 + 10.4403) / 6 ≈ 15.4403 / 6 ≈ 2.5734
- x₂ = (5 - √109) / 6 ≈ (5 - 10.4403) / 6 ≈ -5.4403 / 6 ≈ -0.9067
Rounding these values to three significant figures, we get:
- x₁ ≈ 2.57
- x₂ ≈ -0.907
These are the solutions to the quadratic equation 3x² - 5x - 7 = 0, rounded to three significant figures. The process of rounding is crucial in ensuring the accuracy and practicality of the solutions. Significant figures represent the digits that carry meaning contributing to its precision. In scientific and engineering contexts, it is common to specify the number of significant figures to indicate the level of accuracy required. When rounding, it's important to consider the digit immediately following the last significant figure. If it is 5 or greater, we round up the last significant figure; if it is less than 5, we round down. In this case, we have rounded the fourth significant figure to determine the correct third significant figure. The solutions we have found represent the points where the parabola represented by the equation 3x² - 5x - 7 = 0 intersects the x-axis. These solutions are essential for various applications, including modeling physical phenomena, solving optimization problems, and analyzing data. In the next section, we will summarize our findings and reiterate the steps involved in solving quadratic equations.
Final Answer
In conclusion, to solve the quadratic equation 3x² - 5x - 7 = 0 to three significant figures, we applied the quadratic formula. By identifying the coefficients a = 3, b = -5, and c = -7, we substituted these values into the formula:
x = (-b ± √(b² - 4ac)) / (2a)
This yielded the solutions:
x = (5 ± √109) / 6
Calculating these values and rounding to three significant figures, we obtained the final answers:
- x₁ ≈ 2.57
- x₂ ≈ -0.907
These solutions represent the values of x that satisfy the given quadratic equation. The process of solving quadratic equations using the quadratic formula involves several key steps:
- Identify the coefficients a, b, and c from the equation.
- Substitute these values into the quadratic formula.
- Simplify the expression, including calculating the discriminant.
- Find the two possible solutions by considering both the addition and subtraction of the square root.
- Use a calculator to find the numerical values of the solutions.
- Round the solutions to the required number of significant figures.
This method is universally applicable to any quadratic equation and provides a reliable way to find the roots. Understanding and mastering this process is essential for anyone studying algebra and related fields. The solutions we have found are not just abstract numbers; they have real-world applications in various contexts, from physics and engineering to economics and computer science. By solving quadratic equations, we can model and analyze a wide range of phenomena, making it a fundamental skill in mathematics.
In summary, solving the quadratic equation 3x² - 5x - 7 = 0 to three significant figures involves applying the quadratic formula, calculating the solutions, and rounding appropriately. This process demonstrates the power and versatility of the quadratic formula in solving a fundamental class of mathematical equations. Mastering this skill opens doors to understanding and solving a wide range of problems in various fields. The accurate application of mathematical methods, combined with careful attention to detail, ensures reliable and meaningful results.
