Solving the Quadratic Equation 2X^2 - X - 15 0 Using the Factorization Method

Understanding how to solve quadratic equations is essential for various fields, including mathematical economics. In this article, we will explore the method of solving the equation 2X^2 - X - 15 0 using factorization. Additionally, we will address a common misunderstanding regarding the conditions under which one or both factors can be zero.

Factorizing the Equation

The given equation is:

2X^2 - X - 15 0

To factorize this equation, we need to express it in the form (2x a)(x b) 0. We will solve for a and b such that:

By trial and error, we find that a 5 and b -3 satisfy both conditions. Hence, we can write the equation as:

2X^2 - X - 15 2X^2 - 6X 5X - 15 0

Further factoring gives:

2X(X - 3) 5(X - 3) 0

Or more clearly:

(2X 5)(X - 3) 0

Finding the Solutions

Using the zero property of multiplication, which states that if the product of two numbers is zero, then at least one of the numbers must be zero, we can solve for X as follows:

2X 5 0 Rightarrow X -frac{5}{2}

or

X - 3 0 Rightarrow X 3

Therefore, the solutions are:

X 3 text{ or } X -frac{5}{2}

A Note on Zero Factors

Sometimes, there is confusion about the conditions under which the factors can be zero. Specifically, if ab 0, then either a 0 or b 0, or both. In the case of the factorized equation:

(2X 5)(X - 3) 0

Both factors can be zero simultaneously, meaning either (X -frac{5}{2}) or (X 3) or both can be true.

Conclusion

In summary, the solutions to the quadratic equation 2X^2 - X - 15 0 are X 3 and X -frac{5}{2}, found through the factorization method. Understanding the properties of zero and the conditions under which factors can be zero is crucial for solving such equations accurately.

Additional Resources

For further reading and practice, check out the Quadratic Equation Explanation and Mathway for interactive problem-solving.