Why it is Possible to Have Fractional Solutions in Quadratic Equations Using the Factoring Method

Contrary to the common misconception, it is indeed possible to have fractional solutions when solving a quadratic equation using the factoring method. This article aims to clarify the conditions under which fractions emerge as solutions and provide examples demonstrating how fractional answers are obtained.

Understanding the Factoring Method

The factoring method, also known as the factorization method, is a fundamental technique used to solve quadratic equations of the form ax^2 bx c 0. It involves rewriting the equation such that the left-hand side can be expressed as a product of two binomials.

Example: Solving 2x^2 - 9x 4 0 Using Factoring Method

Consider the quadratic equation: 2x^2 - 9x 4 0. To solve this using the factoring method, we need to factor the quadratic expression on the left-hand side.

Step 1: Identify the coefficients a, b, and c. Here, a 2, b -9, and c 4. Step 2: Look for two numbers that multiply to (2 * 4) 8 and add to -9. These numbers are -8 and -1. Step 3: Rewrite the middle term (-9x) using these numbers: 2x^2 - 8x - x 4 0. Step 4: Factor by grouping: (2x^2 - 8x) - (x - 4) 0. Step 5: Factor out the common terms: 2x(x - 4) - 1(x - 4) 0. Step 6: Combine the terms: (2x - 1)(x - 4) 0.

Setting each factor equal to zero and solving for x, we get:

2x - 1 0 rarr; 2x 1 rarr; x 1/2. x - 4 0 rarr; x 4.

Thus, the solutions to the equation 2x^2 - 9x 4 0 are x 1/2 and x 4, both of which are fractions or whole numbers.

Exploring Further Examples

Let's consider another example to illustrate the possibility of fractional solutions:

Example 1: 4x^2 - 1 0

Factor the equation: (2x 1)(2x - 1) 0. Solve each factor: 2x 1 0 rarr; 2x -1 rarr; x -1/2, and 2x - 1 0 rarr; 2x 1 rarr; x 1/2.

Example 2: 4x^2 - 4x - 3 0

Factor the equation: (2x 1)(2x - 3) 0. Solve each factor: 2x 1 0 rarr; 2x -1 rarr; x -1/2, and 2x - 3 0 rarr; 2x 3 rarr; x 3/2.

These examples demonstrate that the factoring method can result in fractional solutions.

Why Not Have Irrational Solutions?

It's important to note that while fractions are possible, irrational solutions are not achievable through the factoring method alone. If an equation has irrational roots, it cannot be solved by factoring with rational factors. For instance:

Example: 4x^2 - 1 0

Solve each factor: 2x 1 0 rarr; 2x -1 rarr; x -1/2, and 2x - 1 0 rarr; 2x 1 rarr; x 1/2. Here, the roots are rational and can be found by factorization.

For irrational roots, the quadratic formula or completing the square method would be more appropriate techniques to solve the equation.

Conclusion

In conclusion, the factoring method can result in fractional solutions in quadratic equations, as demonstrated in the examples provided. Understanding these solutions helps in solving a wider range of quadratic equations efficiently. Whether the solutions are rational or irrational, the factoring method remains a valuable and useful tool in algebra.