Solving the Quadratic Equation 4x2 - x - 165 0: A Comprehensive Guide

Quadratic equations are a fundamental part of algebra and are encountered in many real-world applications. In this article, we will explore the process of solving the quadratic equation 4x2 - x - 165 0.

Step-by-Step Solution

The given equation is:

4x2 - x - 165 0

Step 1: Prepare the Equation

First, we need to rewrite the equation in standard form:

4x2 - x - 165 0

Move all terms to one side:

4x2 - x - 165 0

Step 2: Solve the Equation

Since the equation is not easily factorizable, we can use the quadratic formula:

x frac{-b pm sqrt{b2 - 4ac}}{2a}

where a 4, b -1, and c -165.

Application of the Quadratic Formula

Substitute the values into the quadratic formula:

x frac{-(-1) pm sqrt{(-1)2 - 4 cdot 4 cdot (-165)}}{2 cdot 4}

Simplify the terms inside the square root:

x frac{1 pm sqrt{1 2640}}{8}

x frac{1 pm sqrt{2641}}{8}

Prime Factorization

To simplify the radical, we can perform prime factorization of 2641:

2641 19 cdot 139

Since 2641 is a product of prime numbers, the square root cannot be simplified further:

x frac{1 pm sqrt{2641}}{8}

Conclusion

The solutions to the equation 4x2 - x - 165 0 are:

x frac{1 sqrt{2641}}{8}

and

x frac{1 - sqrt{2641}}{8}

Further Reading

For a deeper understanding of quadratic equations, you can explore:

Solving quadratic equations by completing the square

Graphical representation of quadratic equations

Applications of quadratic equations in physics and engineering

Understanding these topics will not only enhance your skills in solving such equations but also provide a comprehensive insight into their real-world applications.