Solving Quadratic Polynomials: Finding the Zeros of 3x^2 - 75
Understanding the zeros of a quadratic polynomial is fundamental in algebra and has applications in a variety of fields such as physics, engineering, and economics. In this article, we will explore how to find the zeros of a specific quadratic polynomial, 3x2 - 75, and provide a detailed explanation of the process.
Introduction
A quadratic polynomial is a second-degree polynomial, which means it has the form ax2 bx c. The zeros of a quadratic polynomial are the values of x for which the polynomial equals zero. In the case of 3x2 - 75, the polynomial is already in standard form, and we can directly proceed to find its zeros.
Finding the Zeros
In the given equation, 3x2 - 75 0, we can simplify the process by dividing both sides by 3:
x2 - 25 0
Next, we can factor the equation as follows:
x2 - 25 (x - 5)(x 5) 0
To find the zeros, we set each factor equal to zero:
x - 5 0, x 5 0
Solving these equations, we get:
x 5, x -5
Thus, the zeros of the quadratic polynomial 3x2 - 75 are x 5 and x -5.
Applications and Significance
The zeros of a quadratic polynomial are particularly important because they provide critical information for various applications. In physics, for instance, the zeros can represent the points where an object is at rest (velocity is zero) or changes direction. In engineering, finding the zeros could determine the stress points in a structure or the conditions under which a system operates.
Conclusion
Understanding how to find the zeros of a quadratic polynomial is essential in many areas of mathematics and its applications. By following the steps outlined in this article, you can solve similar problems and gain a deeper understanding of algebraic concepts.
Frequently Asked Questions
Q: What is a quadratic polynomial?
A: A quadratic polynomial is a polynomial of degree two, i.e., it can be written in the form ax2 bx c, where a, b, and c are constants and a ≠ 0.
Q: How many zeros can a quadratic polynomial have?
A: A quadratic polynomial can have up to two distinct zeros. This is because the polynomial can be expressed as a product of two linear factors (if it factors over the real numbers).
Q: Can a quadratic polynomial have complex zeros?
A: Yes, a quadratic polynomial can have complex zeros. This occurs when the discriminant (b2 - 4ac) is negative, and the solutions involve the square root of a negative number, which results in complex numbers.
In summary, finding the zeros of a quadratic polynomial is a fundamental skill that has wide-ranging applications across various fields. By mastering the techniques outlined in this article, you can confidently solve quadratic equations and understand their implications.