Solve The Following Equations: 1) 4x^2 + 3x - 4 = 0 2) 3x^2 + 4x = -10 3) 7X(4 * (-2)) = 0 4) (3x - 4)(2x + 2) = 0
This article delves into solving various algebraic equations, focusing primarily on quadratic equations. We will explore different techniques to find the solutions, also known as roots, for these equations. Let's embark on this mathematical journey and unravel the solutions together.
1) Solving the Quadratic Equation: 4x² + 3x - 4 = 0
In this section, we'll tackle the quadratic equation 4x² + 3x - 4 = 0. Quadratic equations, characterized by the general form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0, are a fundamental part of algebra. To find the values of x that satisfy this equation, we can employ several methods, including factoring, completing the square, or using the quadratic formula. The quadratic formula is a powerful tool that provides the solutions directly, regardless of whether the equation can be factored easily. It is given by:
x = (-b ± √(b² - 4ac)) / (2a)
For our equation, 4x² + 3x - 4 = 0, we can identify a = 4, b = 3, and c = -4. Plugging these values into the quadratic formula, we get:
x = (-3 ± √(3² - 4 * 4 * -4)) / (2 * 4)
Simplifying the expression under the square root:
x = (-3 ± √(9 + 64)) / 8
x = (-3 ± √73) / 8
Thus, we have two solutions:
x₁ = (-3 + √73) / 8
x₂ = (-3 - √73) / 8
These are the roots of the quadratic equation, representing the points where the parabola described by the equation intersects the x-axis. We have successfully used the quadratic formula to solve the equation. This approach is universally applicable to any quadratic equation, making it a valuable technique in algebra. Understanding the quadratic formula and its application is crucial for solving a wide range of mathematical problems. The discriminant (b² - 4ac) within the quadratic formula also gives us important information about the nature of the roots. A positive discriminant indicates two distinct real roots, a zero discriminant indicates one real root (or two identical real roots), and a negative discriminant indicates two complex roots. In our case, the discriminant is 73, which is positive, confirming that we have two distinct real roots. Mastering the quadratic formula empowers students and professionals alike to confidently solve quadratic equations in various contexts, from academic exercises to real-world applications. Remember, the quadratic formula is not just a formula to memorize; it’s a tool to understand and apply. By breaking down the steps and understanding the underlying principles, we can solve quadratic equations effectively and efficiently. This equation demonstrates the power and versatility of algebraic methods in finding solutions to seemingly complex problems.
2) Solving the Quadratic Equation: 3x² + 4x = -10
Let's move on to the next equation: 3x² + 4x = -10. To solve this, we first need to rewrite it in the standard quadratic form, ax² + bx + c = 0. This involves adding 10 to both sides of the equation:
3x² + 4x + 10 = 0
Now we have a quadratic equation with a = 3, b = 4, and c = 10. Again, we can use the quadratic formula to find the solutions:
x = (-b ± √(b² - 4ac)) / (2a)
Substituting the values, we get:
x = (-4 ± √(4² - 4 * 3 * 10)) / (2 * 3)
Simplifying the expression under the square root:
x = (-4 ± √(16 - 120)) / 6
x = (-4 ± √(-104)) / 6
Since the discriminant (the value under the square root) is negative, we have complex solutions. We can rewrite √(-104) as √(104 * -1) = √104 * √-1 = √104 * i, where i is the imaginary unit (√-1). Further simplifying √104, we find that 104 = 4 * 26, so √104 = √(4 * 26) = 2√26. Therefore:
x = (-4 ± 2√26 * i) / 6
We can simplify this further by dividing both the real and imaginary parts by 2:
x = (-2 ± √26 * i) / 3
So, the two complex solutions are:
x₁ = (-2 + √26 * i) / 3
x₂ = (-2 - √26 * i) / 3
These solutions are complex conjugates, meaning they have the same real part but opposite imaginary parts. This is a common occurrence when solving quadratic equations with negative discriminants. The presence of complex roots indicates that the parabola represented by the quadratic equation does not intersect the x-axis. Understanding complex numbers is crucial for solving certain types of quadratic equations and other algebraic problems. Complex numbers extend the number system beyond real numbers and are essential in various fields, including electrical engineering, quantum mechanics, and fluid dynamics. The quadratic formula once again proves its versatility by handling equations with complex solutions. This demonstrates the comprehensive nature of the quadratic formula and its ability to provide solutions regardless of the nature of the roots. In this case, the complex solutions provide valuable information about the behavior of the quadratic function. Equations like this one highlight the importance of understanding the properties of complex numbers in mathematics. These complex solutions remind us that the realm of mathematics extends beyond the real number line, and exploring these concepts enhances our mathematical understanding and problem-solving abilities.
3) Solving the Linear Equation: 7X(4 * (-2)) = 0
Now let's analyze the equation 7X(4 * (-2)) = 0. This is a linear equation, not a quadratic equation, and it can be simplified directly. First, we calculate the value inside the parentheses:
4 * (-2) = -8
So the equation becomes:
7X(-8) = 0
Which simplifies to:
-56X = 0
To solve for X, we divide both sides by -56:
X = 0 / -56
X = 0
Thus, the solution to this equation is X = 0. This is a straightforward linear equation, and its solution is easily obtained by basic algebraic manipulation. Linear equations are fundamental in mathematics and have numerous applications in various fields. They represent a linear relationship between variables and are characterized by having a highest power of 1 for the variable. In this case, the equation represents a line that passes through the origin. Solving linear equations is a cornerstone of algebra and is essential for understanding more complex mathematical concepts. This simple example demonstrates the power of basic algebraic principles in finding solutions. Recognizing the structure of the equation as linear allows for a direct and efficient solution. The solution X = 0 is a critical point, indicating where the line intersects the X-axis. Linear equations form the basis for many mathematical models and are widely used in science, engineering, economics, and other disciplines. Their simplicity and directness make them invaluable tools for problem-solving. This particular equation also serves as a reminder of the importance of order of operations in simplifying expressions. By correctly evaluating the expression inside the parentheses first, we were able to reduce the equation to a simple form that could be easily solved. The solution X = 0 reinforces the concept that zero is a valid solution and plays a significant role in mathematics. This straightforward problem underscores the fundamental principles of algebra and the importance of clear and logical steps in solving equations. Understanding how to solve linear equations is crucial for building a strong foundation in mathematics and for tackling more advanced problems.
4) Solving the Equation by Factoring: (3x - 4)(2x + 2) = 0
Finally, we have the equation (3x - 4)(2x + 2) = 0. This equation is already factored, which makes it easier to solve. The principle we use here is the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for x:
3x - 4 = 0 or 2x + 2 = 0
Solving the first equation, 3x - 4 = 0, we add 4 to both sides:
3x = 4
Then, we divide by 3:
x₁ = 4 / 3
Solving the second equation, 2x + 2 = 0, we subtract 2 from both sides:
2x = -2
Then, we divide by 2:
x₂ = -1
So, the solutions to this equation are x₁ = 4/3 and x₂ = -1. This method of solving equations by factoring is particularly useful when the equation is already in factored form or can be easily factored. Factoring is a powerful technique in algebra that allows us to break down complex expressions into simpler ones. In this case, the equation was already factored, making the application of the zero-product property straightforward. The solutions obtained represent the values of x that make the equation true, or the points where the graph of the equation intersects the x-axis. The zero-product property is a fundamental concept in algebra and is widely used in solving equations of various types. Understanding and applying this property efficiently is essential for success in algebra and related fields. This example highlights the importance of recognizing patterns and structures in equations to choose the most efficient solution method. Factoring is often the quickest way to solve equations when it is applicable. The two distinct solutions obtained indicate that the equation represents a quadratic function that intersects the x-axis at two different points. This reinforces the connection between algebraic solutions and graphical representations of equations. By understanding the principles of factoring and the zero-product property, we can solve a wide range of equations efficiently and accurately. This method provides a clear and direct way to find the solutions, making it an invaluable tool in algebra. This factored form directly lends itself to the zero-product property, offering a clear pathway to the solutions. The ability to recognize and utilize factoring techniques greatly enhances problem-solving skills in algebra.
Conclusion
In this article, we've explored various methods for solving different types of equations, including quadratic and linear equations. We used the quadratic formula to solve quadratic equations, dealt with complex solutions, and applied the zero-product property to solve factored equations. These techniques are fundamental in algebra and provide a solid foundation for more advanced mathematical concepts. By mastering these methods, you'll be well-equipped to tackle a wide range of algebraic problems. The process of solving equations is a cornerstone of mathematics, and the skills acquired in this area are transferable to numerous other disciplines. Each type of equation presents its own challenges and requires a specific approach, highlighting the importance of understanding different techniques. The ability to choose the appropriate method for solving an equation is a key skill that comes with practice and experience. This exploration of various equation-solving techniques underscores the interconnectedness of mathematical concepts and the power of algebraic methods. Understanding the underlying principles behind each method allows for a deeper appreciation of mathematics and its applications in the real world. By building a strong foundation in algebra, we empower ourselves to tackle more complex mathematical problems and to apply these skills in diverse fields. These equation-solving techniques not only help us find numerical solutions but also enhance our logical reasoning and problem-solving abilities, which are valuable assets in any endeavor.