Polynomial Equation With Solution Set {5}
Delving into the realm of polynomial equations, a fundamental concept in algebra, involves identifying equations that possess a specific set of solutions. In this exploration, we embark on a quest to determine the polynomial equation for which the set {5} represents the complete solution set. This means that the value '5' is the only value that, when substituted for the variable 'x' in the equation, will make the equation true. Unraveling this problem requires a deep understanding of polynomial equations, their solutions, and the relationship between them. Let's embark on this mathematical journey to discover the equation that perfectly matches the solution set {5}.
Understanding Polynomial Equations and Solutions
At the heart of this problem lies the concept of polynomial equations. A polynomial equation is a mathematical expression that involves variables raised to non-negative integer powers, combined with constants and mathematical operations such as addition, subtraction, and multiplication. The solutions to a polynomial equation, also known as roots or zeros, are the values of the variable that make the equation true. In simpler terms, they are the values that, when plugged into the equation, result in the equation being equal to zero. The solution set of a polynomial equation is simply the collection of all its solutions.
In our specific case, the solution set is {5}, indicating that the only value that satisfies the polynomial equation is x = 5. This seemingly simple piece of information holds the key to unlocking the equation we seek. To find this equation, we'll leverage the connection between the solutions of a polynomial equation and its factors. Factors are expressions that, when multiplied together, produce the polynomial. Each solution corresponds to a factor of the form (x - solution). Therefore, if 5 is the only solution, then (x - 5) must be a factor of our polynomial. To make it the only solution, we need to ensure this factor appears with a multiplicity of 2, meaning it's squared. This leads us to the equation (x - 5)^2 = 0. Expanding this equation will reveal the polynomial we are looking for. This understanding of the relationship between solutions and factors is crucial in solving this type of problem.
Analyzing the Given Options
Now, let's turn our attention to the provided options, each representing a potential polynomial equation:
- x^2 - 5x + 25 = 0
- x^2 + 10x + 25 = 0
- x^2 - 10x + 25 = 0
- x^2 + 5x + 25 = 0
Our task is to meticulously examine each option to determine which one possesses the solution set {5}. To achieve this, we'll explore two primary methods: the quadratic formula and factoring. The quadratic formula is a powerful tool for finding the solutions of any quadratic equation (an equation of the form ax^2 + bx + c = 0). It provides a direct way to calculate the solutions based on the coefficients a, b, and c. Alternatively, factoring involves breaking down the quadratic expression into the product of two linear expressions. If we can factor an equation into the form (x - 5)(x - 5) = 0, we've found our answer. By applying these methods to each option, we'll systematically narrow down the possibilities and identify the correct polynomial equation.
Applying the Quadratic Formula
The quadratic formula is a reliable method for solving quadratic equations of the form ax^2 + bx + c = 0. The formula is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
To use the quadratic formula, we simply substitute the coefficients a, b, and c from the quadratic equation into the formula and simplify. The result will be the solutions to the equation. If the discriminant (b^2 - 4ac) is positive, there are two distinct real solutions. If it's zero, there is exactly one real solution (a repeated root). If it's negative, there are two complex solutions. For our problem, we are looking for an equation with the single real solution x = 5. This means the discriminant must be zero, and the quadratic formula should yield x = 5 as the only solution. We will apply this formula to each of the options, carefully calculating the discriminant and the resulting solutions to see which one matches our criteria.
Factoring the Quadratic Equations
Factoring is an alternative method for solving quadratic equations. It involves expressing the quadratic expression as the product of two linear factors. For example, the expression x^2 - 4 can be factored as (x - 2)(x + 2). If we can factor a quadratic equation into the form (x - 5)(x - 5) = 0, we know that x = 5 is the only solution. This is because setting each factor to zero (x - 5 = 0) yields the same solution, x = 5. Factoring can be a quicker method than the quadratic formula, especially when the quadratic expression has integer roots. However, not all quadratic expressions can be easily factored. In such cases, the quadratic formula is a more reliable approach. We will attempt to factor each of the given options to see if we can find one that factors into the desired form, (x - 5)(x - 5) = 0.
Step-by-Step Solution
Let's systematically analyze each option:
Option 1: x^2 - 5x + 25 = 0
Using the quadratic formula:
x = (5 ± √((-5)^2 - 4 * 1 * 25)) / (2 * 1)
x = (5 ± √(25 - 100)) / 2
x = (5 ± √(-75)) / 2
The discriminant is negative, indicating complex solutions. Therefore, this option is incorrect.
Option 2: x^2 + 10x + 25 = 0
Attempting to factor:
(x + 5)(x + 5) = 0
This gives us x = -5 as the solution. Therefore, this option is incorrect.
Option 3: x^2 - 10x + 25 = 0
Attempting to factor:
(x - 5)(x - 5) = 0
This gives us x = 5 as the only solution. This option matches our criteria!
Option 4: x^2 + 5x + 25 = 0
Using the quadratic formula:
x = (-5 ± √(5^2 - 4 * 1 * 25)) / (2 * 1)
x = (-5 ± √(25 - 100)) / 2
x = (-5 ± √(-75)) / 2
The discriminant is negative, indicating complex solutions. Therefore, this option is incorrect.
The Correct Polynomial Equation
Through our step-by-step analysis, we have identified that x^2 - 10x + 25 = 0 is the polynomial equation with the solution set {5}. This equation can be factored into (x - 5)(x - 5) = 0, which clearly demonstrates that x = 5 is the only solution. The other options either had complex solutions or a different solution set. Therefore, option 3 is the definitive answer to our problem. This exploration has not only provided the solution but also reinforced our understanding of the relationship between polynomial equations, their solutions, and factoring techniques.
Conclusion: Mastering Polynomial Equations
In this mathematical endeavor, we successfully unveiled the polynomial equation possessing the solution set {5}. We navigated through the intricacies of polynomial equations, factoring, and the quadratic formula, ultimately pinpointing x^2 - 10x + 25 = 0 as the definitive solution. This exercise serves as a testament to the power of analytical thinking and the application of fundamental algebraic principles. By grasping the connection between solutions and factors, and by mastering techniques like the quadratic formula, we can confidently tackle a wide range of polynomial equation problems. This journey into the world of polynomials has not only provided a solution but has also enhanced our problem-solving skills and deepened our appreciation for the elegance of mathematics.
