Finding The Quadratic Equation With Solutions X = 1 ± √5

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Finding the correct quadratic equation that has the solutions x = 1 ± √5 involves understanding the relationship between the roots of a quadratic equation and its coefficients. This article will walk you through the process of identifying the correct equation, explaining the underlying concepts, and providing a step-by-step solution. Whether you're a student grappling with algebra or just looking to refresh your math skills, this guide will help you master this type of problem.

Understanding Quadratic Equations and Their Roots

Before diving into the specific problem, let's recap some fundamental concepts about quadratic equations. A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to this equation, also known as roots or zeros, are the values of x that satisfy the equation. Quadratic equations can have two distinct real roots, one repeated real root, or two complex roots.

The roots of a quadratic equation can be found using the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

This formula is derived from the method of completing the square and is a powerful tool for solving any quadratic equation. The expression b² - 4ac under the square root is called the discriminant, and it provides valuable information about the nature of the roots:

  • If b² - 4ac > 0, the equation has two distinct real roots.
  • If b² - 4ac = 0, the equation has one repeated real root.
  • If b² - 4ac < 0, the equation has two complex roots.

Another crucial concept is the relationship between the roots and the coefficients of the quadratic equation. If x₁ and x₂ are the roots of the quadratic equation ax² + bx + c = 0, then:

  • Sum of the roots: x₁ + x₂ = -b/a
  • Product of the roots: x₁ * x₂ = c/a

These relationships are incredibly useful for constructing a quadratic equation given its roots, or for verifying whether a given set of roots corresponds to a particular equation. In this article, we will primarily use these relationships to find the correct quadratic equation.

Step-by-Step Solution: Finding the Correct Equation

Now, let's apply these concepts to the given problem. We are looking for the quadratic equation that has the solutions x = 1 ± √5. This means we have two roots:

  • x₁ = 1 + √5
  • x₂ = 1 - √5

To find the quadratic equation, we can use the relationships between the roots and the coefficients. First, let's find the sum of the roots:

x₁ + x₂ = (1 + √5) + (1 - √5) = 1 + √5 + 1 - √5 = 2

Next, let's find the product of the roots:

x₁ * x₂ = (1 + √5)(1 - √5) = 1 - 5 = -4

Now, we know that for a quadratic equation of the form ax² + bx + c = 0, the sum of the roots is -b/a and the product of the roots is c/a. If we assume a = 1 (which simplifies the equation without loss of generality), then:

  • -b/1 = 2, so b = -2
  • c/1 = -4, so c = -4

Therefore, the quadratic equation is:

x² - 2x - 4 = 0

Verification

To verify that this is indeed the correct equation, we can use the quadratic formula to solve x² - 2x - 4 = 0:

x = (-(-2) ± √((-2)² - 4(1)(-4))) / (2(1)) x = (2 ± √(4 + 16)) / 2 x = (2 ± √20) / 2 x = (2 ± 2√5) / 2 x = 1 ± √5

This confirms that the solutions to the equation x² - 2x - 4 = 0 are indeed x = 1 ± √5. Therefore, the correct quadratic equation is x² - 2x - 4 = 0. This step-by-step verification underscores the accuracy of our method and the correctness of the solution.

Analyzing the Incorrect Options

To further solidify our understanding, let's briefly analyze why the other options are incorrect. This will not only confirm our answer but also deepen our grasp of quadratic equations and their roots.

  1. x² + 2x - 4 = 0:
    • In this equation, a = 1, b = 2, and c = -4. The sum of the roots would be -b/a = -2, and the product of the roots would be c/a = -4. While the product of the roots matches our calculated product (-4), the sum of the roots does not match our calculated sum (2). Therefore, this equation cannot have the solutions x = 1 ± √5.
  2. x² - 2x + 4 = 0:
    • Here, a = 1, b = -2, and c = 4. The sum of the roots is -b/a = 2, which matches our calculated sum. However, the product of the roots is c/a = 4, which does not match our calculated product (-4). Thus, this equation is also incorrect.
  3. x² + 2x + 4 = 0:
    • In this case, a = 1, b = 2, and c = 4. The sum of the roots is -b/a = -2, which does not match our calculated sum (2), and the product of the roots is c/a = 4, which does not match our calculated product (-4). This equation does not have the given solutions.

By analyzing these incorrect options, we reinforce the understanding that both the sum and the product of the roots must match the values derived from the coefficients of the quadratic equation. This comprehensive analysis provides a strong understanding of why the other equations are not viable solutions.

Alternative Method: Constructing the Equation Directly

There's another method we can use to construct the quadratic equation directly from its roots. This method involves using the fact that if x₁ and x₂ are the roots of a quadratic equation, then the equation can be written in the form:

(x - x₁)(x - x₂) = 0

In our case, x₁ = 1 + √5 and x₂ = 1 - √5. Substituting these values into the equation, we get:

(x - (1 + √5))(x - (1 - √5)) = 0

Expanding this, we have:

x² - x(1 - √5) - x(1 + √5) + (1 + √5)(1 - √5) = 0 x² - x + x√5 - x - x√5 + (1 - 5) = 0 x² - 2x - 4 = 0

This method directly gives us the quadratic equation x² - 2x - 4 = 0, confirming our earlier result. This alternative method highlights the versatility in solving quadratic equation problems and provides another tool for verification.

Benefits of Knowing Multiple Methods

Understanding multiple methods to solve a problem is extremely beneficial for several reasons. Firstly, it allows for verification of results. If you arrive at the same answer using different approaches, you can be more confident in the correctness of your solution. Secondly, different methods may be more suitable for different problems, or for different people depending on their preferred style of thinking. Knowing multiple methods enhances your problem-solving skills and gives you the flexibility to choose the most efficient approach.

Practical Applications and Real-World Examples

While solving quadratic equations might seem like a purely academic exercise, quadratic equations have numerous practical applications in various fields. Understanding these applications can provide a deeper appreciation for the importance of this mathematical concept. Quadratic equations are used extensively in physics, engineering, economics, and computer science.

Physics

In physics, quadratic equations are used to describe projectile motion. For example, the height of a projectile (like a ball thrown into the air) as a function of time can be modeled using a quadratic equation. The roots of the equation can tell us when the projectile hits the ground. Understanding quadratic equations is crucial for predicting the trajectory and range of projectiles.

Engineering

Engineers use quadratic equations in designing structures, electrical circuits, and control systems. For instance, in electrical engineering, the current in a circuit containing resistors, inductors, and capacitors can often be described using quadratic equations. These equations help engineers optimize designs and ensure the stability and efficiency of various systems.

Economics

In economics, quadratic functions are used to model cost, revenue, and profit functions. By finding the roots and vertex of a quadratic function, economists can determine break-even points and maximize profits. Quadratic equations are essential tools for economic analysis and decision-making.

Computer Science

Quadratic equations also appear in computer science, particularly in areas like computer graphics and optimization algorithms. For example, quadratic Bézier curves are used to create smooth curves in computer graphics, and the solutions of quadratic equations can help optimize the performance of algorithms. These applications demonstrate the broad utility of quadratic equations in the digital world.

Real-World Examples

Consider a simple real-world example: maximizing the area of a rectangular garden with a fixed perimeter. If you have a certain amount of fencing material, the area of the garden can be expressed as a quadratic function of its dimensions. By solving the related quadratic equation, you can determine the dimensions that will maximize the garden's area. This illustrates how quadratic equations can help solve practical optimization problems in everyday life.

Conclusion: Mastering Quadratic Equations

In conclusion, finding the quadratic equation with solutions x = 1 ± √5 involves understanding the relationship between the roots and coefficients of quadratic equations. By calculating the sum and product of the roots, we identified that the correct equation is x² - 2x - 4 = 0. We verified this result using the quadratic formula and by constructing the equation directly from its roots. Analyzing the incorrect options further solidified our understanding and highlighted the importance of both the sum and product of the roots.

We also explored practical applications of quadratic equations in various fields, demonstrating their relevance beyond academic exercises. Mastering quadratic equations provides a valuable toolset for problem-solving in physics, engineering, economics, computer science, and many other areas.

By understanding the concepts discussed in this article and practicing similar problems, you can strengthen your algebraic skills and approach quadratic equations with confidence. This comprehensive guide aims to equip you with the knowledge and skills necessary to excel in solving quadratic equations and appreciate their broad applicability. Whether you are a student, professional, or simply someone interested in mathematics, understanding quadratic equations is a valuable asset.