Finding The Roots Of The Quadratic Function F(b) = B^2 - 75
In this article, we will delve into the process of finding the roots of the quadratic function f(b) = b^2 - 75. This is a fundamental concept in algebra, and understanding how to solve such equations is crucial for various mathematical applications. We will explore the methods to identify the roots, which are the values of b that make the function equal to zero. By the end of this discussion, you will have a clear understanding of how to solve quadratic equations of this form and be able to apply this knowledge to similar problems.
Understanding Quadratic Functions
Quadratic functions are polynomial functions of the second degree. The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants, and a is not equal to zero. The graph of a quadratic function is a parabola, a U-shaped curve that opens upwards if a > 0 and downwards if a < 0. The roots of a quadratic function, also known as the zeros or solutions, are the values of x for which the function f(x) equals zero. These roots represent the points where the parabola intersects the x-axis.
In our specific case, the quadratic function is f(b) = b^2 - 75. This is a simplified form where the coefficient of the b term is zero (i.e., b = 0 in the general form). This simplification makes the process of finding the roots more straightforward. The roots of this function will give us the values of b that satisfy the equation b^2 - 75 = 0. Understanding the nature of quadratic functions and their graphical representation helps in visualizing the solutions and provides a solid foundation for solving more complex problems.
Methods to Find the Roots
There are several methods to find the roots of a quadratic equation, but for the function f(b) = b^2 - 75, the most straightforward approach is to use the square root property. This method is particularly effective when the quadratic equation is in the form x^2 = k, where k is a constant. In this case, the roots can be found by taking the square root of both sides of the equation.
Another common method is factoring, but it is more suitable for quadratic equations that can be easily factored into two binomials. The quadratic formula is a more general method that can be used for any quadratic equation, but it is often more complex to apply than the square root property for simpler equations like ours. Completing the square is another method, but it is typically used when the quadratic formula is not readily applicable or when converting the quadratic equation into vertex form.
For our function, using the square root property is the most efficient way to find the roots. We will set the function equal to zero and then isolate b^2 to one side of the equation. From there, we will take the square root of both sides, remembering to consider both the positive and negative roots. This method will directly lead us to the solutions for b that satisfy the equation b^2 - 75 = 0.
Applying the Square Root Property
To find the roots of the quadratic function f(b) = b^2 - 75, we first set the function equal to zero:
b^2 - 75 = 0
Next, we isolate the b^2 term by adding 75 to both sides of the equation:
b^2 = 75
Now, we apply the square root property by taking the square root of both sides. It's crucial to remember that when taking the square root, we must consider both the positive and negative roots:
b = ±√75
To simplify the square root, we look for perfect square factors of 75. We can express 75 as 25 * 3, where 25 is a perfect square. Therefore, we can rewrite the equation as:
b = ±√(25 * 3)
Using the property of square roots that √(a * b) = √a * √b, we can further simplify:
b = ±√25 * √3
Since √25 = 5, the equation becomes:
b = ±5√3
Thus, the roots of the quadratic function f(b) = b^2 - 75 are b = 5√3 and b = -5√3. These are the two values of b that make the function equal to zero.
Verifying the Roots
To ensure the accuracy of our solution, it's always a good practice to verify the roots by substituting them back into the original equation. Let's verify the roots b = 5√3 and b = -5√3 in the function f(b) = b^2 - 75.
First, let's substitute b = 5√3:
f(5√3) = (5√3)^2 - 75
Squaring 5√3 means squaring both the 5 and the √3:
(5√3)^2 = 5^2 * (√3)^2 = 25 * 3 = 75
So, the equation becomes:
f(5√3) = 75 - 75 = 0
This confirms that b = 5√3 is indeed a root of the function.
Now, let's substitute b = -5√3:
f(-5√3) = (-5√3)^2 - 75
Again, squaring -5√3 means squaring both the -5 and the √3:
(-5√3)^2 = (-5)^2 * (√3)^2 = 25 * 3 = 75
So, the equation becomes:
f(-5√3) = 75 - 75 = 0
This confirms that b = -5√3 is also a root of the function. Therefore, our solutions are correct.
Conclusion
In summary, we have successfully found the roots of the quadratic function f(b) = b^2 - 75 using the square root property. The roots are b = 5√3 and b = -5√3. We verified these roots by substituting them back into the original equation and confirming that they make the function equal to zero. This exercise demonstrates the importance of understanding quadratic functions and the various methods available to solve them. The square root property is a powerful tool for solving equations of the form x^2 = k, and it provides a direct and efficient way to find the roots.
By mastering these fundamental concepts, you will be well-equipped to tackle more complex algebraic problems and gain a deeper understanding of mathematical principles. Remember, practice is key to proficiency, so continue to explore different types of quadratic equations and apply these methods to build your problem-solving skills. Understanding the roots of quadratic functions is not only essential for algebra but also lays the foundation for more advanced mathematical topics such as calculus and complex analysis.
