Solution Set Of X² + 5x - 5 = 0 A Step By Step Guide
In the realm of mathematics, quadratic equations hold a prominent position. These equations, characterized by the highest power of the variable being 2, appear frequently in various mathematical contexts and real-world applications. One such equation is x² + 5x - 5 = 0. Our goal here is to unravel the mystery behind this equation and determine its solution set, which represents the values of x that satisfy the equation.
Understanding Quadratic Equations
Before diving into the specifics of our equation, let's 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 general form of a quadratic equation is:
ax² + bx + c = 0
where a, b, and c are constants, and a is not equal to 0 (otherwise, it would be a linear equation). The coefficients a, b, and c play crucial roles in determining the nature and solutions of the quadratic equation.
Quadratic equations can have up to two distinct solutions, also known as roots or zeros. These solutions represent the points where the quadratic function intersects the x-axis when graphed. Finding these solutions is a fundamental task in algebra, and several methods exist to accomplish this.
Methods for Solving Quadratic Equations
Several techniques are available for solving quadratic equations, each with its strengths and suitability for different types of equations. The most common methods include:
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Factoring: This method involves expressing the quadratic expression as a product of two linear factors. If the equation can be factored, the solutions can be easily found by setting each factor equal to zero and solving for x.
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Completing the Square: This technique involves manipulating the equation to create a perfect square trinomial on one side. This allows us to take the square root of both sides and solve for x.
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Quadratic Formula: This formula provides a direct solution for any quadratic equation, regardless of whether it can be factored or not. The quadratic formula is given by:
x = (-b ± √(b² - 4ac)) / 2a
where a, b, and c are the coefficients of the quadratic equation.
The choice of method depends on the specific equation and the solver's preference. For some equations, factoring might be the quickest approach, while for others, the quadratic formula might be the most reliable.
Applying the Quadratic Formula to x² + 5x - 5 = 0
In our case, the equation x² + 5x - 5 = 0 does not lend itself easily to factoring. Therefore, we will employ the quadratic formula to find its solutions. Comparing our equation to the general form ax² + bx + c = 0, we identify the coefficients as:
- a = 1
- b = 5
- c = -5
Now, we substitute these values into the quadratic formula:
x = (-5 ± √(5² - 4 * 1 * -5)) / (2 * 1)
Simplifying the expression under the square root:
x = (-5 ± √(25 + 20)) / 2
x = (-5 ± √45) / 2
We can further simplify √45 by factoring out the largest perfect square:
x = (-5 ± √(9 * 5)) / 2
x = (-5 ± 3√5) / 2
Thus, we arrive at two distinct solutions:
x₁ = (-5 + 3√5) / 2
x₂ = (-5 - 3√5) / 2
These values represent the solution set of the equation x² + 5x - 5 = 0.
The Solution Set and Its Significance
The solution set of a quadratic equation is the set of all values of the variable that satisfy the equation. In our case, the solution set consists of two real numbers:
{(-5 + 3√5) / 2, (-5 - 3√5) / 2}
These solutions represent the x-intercepts of the parabola defined by the quadratic function y = x² + 5x - 5. The parabola intersects the x-axis at these two points.
The solutions of a quadratic equation are crucial in various mathematical and real-world applications. They can represent the roots of a function, the points of intersection of two curves, or the values that satisfy a particular condition in a problem. Understanding the solution set provides valuable insights into the behavior of the quadratic equation and its related functions.
Exploring the Discriminant
An important part of the quadratic formula is the discriminant, which is the expression under the square root:
Δ = b² - 4ac
The discriminant provides valuable information about the nature of the solutions of a quadratic equation. There are three possible cases:
- If Δ > 0, the equation has two distinct real solutions.
- If Δ = 0, the equation has one real solution (a repeated root).
- If Δ < 0, the equation has two complex solutions.
In our example, the discriminant is:
Δ = 5² - 4 * 1 * -5 = 25 + 20 = 45
Since Δ > 0, we confirm that the equation x² + 5x - 5 = 0 has two distinct real solutions, as we found earlier.
Graphical Interpretation
The solutions of a quadratic equation can be visualized graphically. The graph of the quadratic function y = ax² + bx + c is a parabola. The x-intercepts of the parabola correspond to the solutions of the equation ax² + bx + c = 0.
In the case of x² + 5x - 5 = 0, the graph is a parabola that opens upwards (since a = 1 is positive) and intersects the x-axis at the two points corresponding to the solutions we found: (-5 + 3√5) / 2 and (-5 - 3√5) / 2.
Visualizing the graph provides a geometric understanding of the solutions and their relationship to the quadratic function.
Applications of Quadratic Equations
Quadratic equations have numerous applications in various fields, including:
- Physics: Projectile motion, the trajectory of objects under gravity, can be modeled using quadratic equations.
- Engineering: Designing structures, bridges, and other constructions often involves solving quadratic equations.
- Economics: Modeling supply and demand curves, profit maximization, and cost minimization can utilize quadratic equations.
- Computer Graphics: Quadratic equations are used to create curves and surfaces in computer graphics and animation.
The versatility of quadratic equations makes them a fundamental tool in problem-solving across diverse disciplines.
Conclusion
In this comprehensive exploration, we successfully determined the solution set of the quadratic equation x² + 5x - 5 = 0. We employed the quadratic formula, a powerful tool for solving any quadratic equation. We found that the equation has two distinct real solutions:
x₁ = (-5 + 3√5) / 2
x₂ = (-5 - 3√5) / 2
These solutions represent the values of x that satisfy the equation and correspond to the x-intercepts of the parabola defined by the quadratic function. We also discussed the significance of the discriminant in determining the nature of the solutions and explored the graphical interpretation of quadratic equations.
Quadratic equations are fundamental mathematical tools with wide-ranging applications. Understanding how to solve them and interpret their solutions is essential for anyone working in mathematics, science, engineering, or related fields. This exploration has provided a deep dive into the solution of a specific quadratic equation, highlighting the key concepts and techniques involved in this important area of mathematics.
This comprehensive guide delves into the realm of quadratic equations, focusing specifically on how to determine the solution set for the equation x² + 5x - 5 = 0. Quadratic equations, a cornerstone of algebra, are polynomial equations of the second degree, characterized by the presence of a term where the variable is raised to the power of two. These equations manifest themselves in numerous mathematical contexts and real-world scenarios, making their understanding crucial for anyone venturing into fields like physics, engineering, economics, and computer science.
What are Quadratic Equations?
At their core, quadratic equations are algebraic expressions that can be written in the general form:
ax² + bx + c = 0
Where a, b, and c are constants, with a not equaling zero (as this would reduce the equation to a linear one). The coefficients a, b, and c hold the key to understanding the equation's properties and finding its solutions. The solution set for a quadratic equation is the set of values for x that make the equation true.
These solutions, also known as roots or zeros, represent the points where the parabola described by the quadratic function intersects the x-axis when graphed. A quadratic equation can have two, one, or no real solutions, depending on the values of its coefficients.
Methods for Unraveling Quadratic Equations
There are several established methods for solving quadratic equations, each with its own advantages and suitability depending on the specific equation. The most common methods include:
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Factoring: This elegant method involves rewriting the quadratic expression as a product of two linear factors. If successful, this allows the solutions to be found by setting each factor equal to zero and solving for x. Factoring is generally the quickest method when it's applicable, but not all quadratic equations can be easily factored.
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Completing the Square: This technique involves algebraic manipulation to transform the equation into a perfect square trinomial on one side. By taking the square root of both sides, the equation can be solved for x. Completing the square is a reliable method, particularly useful for understanding the derivation of the quadratic formula and for cases where factoring is not straightforward.
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The Quadratic Formula: This universally applicable formula provides a direct solution for any quadratic equation, regardless of its factorability. Derived by completing the square on the general form of the quadratic equation, the quadratic formula is expressed as:
x = (-b ± √(b² - 4ac)) / 2a
Where a, b, and c are the coefficients of the quadratic equation. The quadratic formula guarantees a solution and is often the go-to method for complex or unfamiliar equations.
The choice of method is often determined by the equation's specific form and the solver's personal preference. Simpler equations might yield quickly to factoring, while more complex ones might necessitate the use of the quadratic formula.
Cracking the Code: Solving x² + 5x - 5 = 0 Using the Quadratic Formula
In our case, the equation x² + 5x - 5 = 0 presents a challenge for factoring due to its coefficients. Therefore, we'll leverage the power of the quadratic formula to determine its solutions. By comparing our equation to the general form ax² + bx + c = 0, we can identify the coefficients as follows:
- a = 1 (the coefficient of x²)
- b = 5 (the coefficient of x)
- c = -5 (the constant term)
Now, we substitute these values into the quadratic formula:
x = (-5 ± √(5² - 4 * 1 * -5)) / (2 * 1)
Let's simplify this step-by-step. First, we address the expression under the square root:
x = (-5 ± √(25 + 20)) / 2
x = (-5 ± √45) / 2
We can further simplify √45 by factoring out the largest perfect square it contains:
x = (-5 ± √(9 * 5)) / 2
x = (-5 ± 3√5) / 2
This yields two distinct solutions for x:
x₁ = (-5 + 3√5) / 2
x₂ = (-5 - 3√5) / 2
Therefore, the solution set for the equation x² + 5x - 5 = 0 is the set containing these two real numbers.
The Significance of the Solution Set
The solution set of a quadratic equation is more than just a set of numbers; it represents the values of the variable that make the equation a true statement. In our specific case, the solution set:
{(-5 + 3√5) / 2, (-5 - 3√5) / 2}
Consists of two real numbers. These solutions hold a significant geometric interpretation: they represent the x-intercepts of the parabola defined by the quadratic function y = x² + 5x - 5. At these points, the parabola crosses the x-axis.
Understanding the solutions of a quadratic equation is vital in numerous applications. They can represent the points where a projectile hits the ground, the dimensions of a rectangle with a specific area, or the equilibrium points in a dynamic system. The solution set provides crucial insights into the behavior and characteristics of the quadratic equation and its related functions.
Decoding the Discriminant: Unveiling the Nature of Solutions
A key component of the quadratic formula is the discriminant, represented by the expression under the square root:
Δ = b² - 4ac
The discriminant serves as a powerful indicator of the nature of the solutions a quadratic equation possesses. It reveals whether the solutions are real and distinct, real and repeated, or complex. There are three possible scenarios:
- Δ > 0: The equation has two distinct real solutions. This means the parabola intersects the x-axis at two different points.
- Δ = 0: The equation has one real solution (a repeated root). In this case, the parabola touches the x-axis at only one point.
- Δ < 0: The equation has two complex solutions (conjugate pairs). This indicates that the parabola does not intersect the x-axis.
For our equation, x² + 5x - 5 = 0, the discriminant is:
Δ = 5² - 4 * 1 * -5 = 25 + 20 = 45
Since Δ > 0, we can confirm that our equation indeed has two distinct real solutions, as we determined earlier using the quadratic formula.
Visualizing the Solutions: The Graphical Perspective
Quadratic equations have a strong visual representation. The graph of a quadratic function y = ax² + bx + c is a parabola, a U-shaped curve. The solutions of the equation ax² + bx + c = 0 correspond to the x-intercepts of this parabola.
In the case of x² + 5x - 5 = 0, the graph is an upward-opening parabola (because a = 1 is positive), intersecting the x-axis at the two points corresponding to our solutions: (-5 + 3√5) / 2 and (-5 - 3√5) / 2.
Visualizing the parabola provides a geometric understanding of the solutions and their relationship to the quadratic function. The x-intercepts represent the points where the function's value is zero.
Applications Across Disciplines: The Pervasive Nature of Quadratic Equations
Quadratic equations are not confined to the realm of pure mathematics; they appear extensively in various real-world applications, including:
- Physics: Modeling projectile motion, calculating the trajectory of objects under the influence of gravity.
- Engineering: Designing bridges, buildings, and other structures, ensuring stability and load-bearing capacity.
- Economics: Modeling supply and demand curves, optimizing production and pricing strategies.
- Computer Graphics: Creating curves and surfaces for 3D modeling, animation, and game development.
The broad applicability of quadratic equations underscores their importance as a fundamental tool for problem-solving across diverse fields.
Conclusion: Mastering Quadratic Equations
In this in-depth exploration, we've successfully navigated the process of finding the solution set for the quadratic equation x² + 5x - 5 = 0. We've employed the quadratic formula, a powerful technique that guarantees a solution for any quadratic equation. Our analysis revealed two distinct real solutions:
x₁ = (-5 + 3√5) / 2
x₂ = (-5 - 3√5) / 2
These solutions represent the values of x that satisfy the equation, corresponding to the x-intercepts of the associated parabola. We've also delved into the significance of the discriminant, providing insights into the nature of the solutions, and explored the graphical representation of quadratic equations.
Quadratic equations are indispensable mathematical tools with far-reaching applications. A solid understanding of how to solve them and interpret their solutions is crucial for anyone pursuing studies or careers in science, technology, engineering, and mathematics (STEM) fields. This guide has provided a comprehensive overview of the key concepts and techniques involved in unraveling quadratic equations, empowering you to confidently tackle these fundamental mathematical challenges.
Quadratic equations, a fundamental part of algebra, show up in many areas of math and science. Knowing how to solve them is a key skill. In this article, we'll explore how to find the solution set for the equation x² + 5x - 5 = 0. We'll use different methods and explain why they work. Whether you're a student learning algebra or just looking to brush up on your math skills, this guide will help you understand how to tackle quadratic equations.
What Makes an Equation Quadratic?
A quadratic equation is any equation that can be written in this general form:
ax² + bx + c = 0
Here, x is the variable we want to solve for, and a, b, and c are constants, with a not equal to zero. The term ax² is what makes the equation quadratic; it's the highest power of x, which is 2. The values of a, b, and c determine the shape and position of the parabola that the equation represents when graphed. The solution set of a quadratic equation is the set of x values that make the equation true.
These solutions are also called roots or zeros. They are the points where the parabola crosses the x-axis. A quadratic equation can have two, one, or no real solutions. The number of solutions depends on the values of a, b, and c.
Different Ways to Solve Quadratic Equations
There are several methods for solving quadratic equations, each with its own advantages. The most common ones are:
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Factoring: If we can write the quadratic expression ax² + bx + c as a product of two linear factors, we can find the solutions easily. We set each factor equal to zero and solve for x. Factoring is often the quickest method when it works, but it's not always easy to factor a quadratic expression.
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Completing the Square: This method involves rewriting the equation so that one side is a perfect square trinomial. This allows us to take the square root of both sides and solve for x. Completing the square is a reliable method that works for all quadratic equations. It's also useful for understanding the derivation of the quadratic formula.
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The Quadratic Formula: This formula gives us the solutions directly, no matter how complex the equation is. It's derived by completing the square on the general form of the quadratic equation. The formula is:
x = (-b ± √(b² - 4ac)) / 2a
Here, a, b, and c are the coefficients from the quadratic equation ax² + bx + c = 0. The quadratic formula is a powerful tool, especially when factoring is difficult or impossible.
The method we choose depends on the equation itself and our personal preference. Some equations are easily factored, while others require the quadratic formula.
Solving x² + 5x - 5 = 0 Using the Quadratic Formula
For our equation, x² + 5x - 5 = 0, factoring isn't straightforward. So, we'll use the quadratic formula. First, we identify the coefficients:
- a = 1
- b = 5
- c = -5
Now, we plug these values into the formula:
x = (-5 ± √(5² - 4 * 1 * -5)) / (2 * 1)
Let's simplify step by step:
x = (-5 ± √(25 + 20)) / 2
x = (-5 ± √45) / 2
We can simplify √45 further by factoring out the largest perfect square:
x = (-5 ± √(9 * 5)) / 2
x = (-5 ± 3√5) / 2
This gives us two solutions:
x₁ = (-5 + 3√5) / 2
x₂ = (-5 - 3√5) / 2
So, the solution set for x² + 5x - 5 = 0 is the set containing these two values.
What the Solutions Tell Us
The solution set is more than just a pair of numbers; it tells us where the quadratic function y = x² + 5x - 5 crosses the x-axis. These points are the x-intercepts of the parabola. In our case, the solutions are real numbers, which means the parabola intersects the x-axis at two distinct points. If the solutions were complex numbers, the parabola wouldn't intersect the x-axis at all.
Understanding the solutions helps us visualize the quadratic function and its behavior. It also allows us to solve real-world problems that can be modeled by quadratic equations.
The Discriminant: A Key to Understanding Solutions
Inside the quadratic formula, there's a special part called the discriminant. It's the expression under the square root:
Δ = b² - 4ac
The discriminant tells us a lot about the nature of the solutions:
- If Δ > 0, there are two distinct real solutions.
- If Δ = 0, there is one real solution (a repeated root).
- If Δ < 0, there are two complex solutions.
For our equation, the discriminant is:
Δ = 5² - 4 * 1 * -5 = 25 + 20 = 45
Since Δ > 0, we know that there are two distinct real solutions, which confirms what we found earlier.
Seeing the Solutions: The Graph of the Equation
We can visualize the solutions by graphing the quadratic function y = x² + 5x - 5. The graph is a parabola that opens upwards because the coefficient of x² is positive. The points where the parabola crosses the x-axis are the solutions to the equation. In our case, the parabola intersects the x-axis at approximately x = 0.85 and x = -5.85, which correspond to our solutions (-5 + 3√5) / 2 and (-5 - 3√5) / 2.
Graphing the equation provides a visual confirmation of our solutions and helps us understand the relationship between the equation and its roots.
Where Quadratic Equations Are Used
Quadratic equations aren't just abstract math; they have many practical uses in various fields:
- Physics: Modeling projectile motion, calculating the range and height of a thrown object.
- Engineering: Designing structures, determining the optimal shape for arches and bridges.
- Economics: Modeling supply and demand curves, finding equilibrium points.
- Computer Graphics: Creating curves and surfaces for 3D modeling and animation.
The wide range of applications highlights the importance of understanding and solving quadratic equations.
Final Thoughts
In this guide, we've thoroughly explored how to find the solution set for the quadratic equation x² + 5x - 5 = 0. We used the quadratic formula, explained why it works, and discussed the significance of the solutions and the discriminant. We also looked at the graphical representation of the equation and its applications in real-world scenarios.
By mastering quadratic equations, you're not just learning a math skill; you're gaining a powerful tool for solving problems in various fields. Whether you're a student, a professional, or simply someone who enjoys math, understanding quadratic equations is a valuable asset in the world of mathematics and beyond.
