Solving Quadratic Equations with Fractions: A Comprehensive Guide

Quadratic equations are fundamental in algebra and often involve fractions, which can make solving them more challenging. This article will guide you through the process of solving a quadratic equation with fractions and why simply multiplying by the denominator to eliminate the fractions is not a valid method. We will delve into the quadratic formula and provide a detailed step-by-step solution for a given example.

Understanding Quadratic Equations

A quadratic equation is a second-degree polynomial equation of the form (ax^2 bx c 0), where (a), (b), and (c) are constants, and (a eq 0). When dealing with fractions, the process is slightly more complex, but the principles remain the same.

The Process of Solving a Quadratic Equation with Fractions

Let's consider the following example equation:

x2   (1/4)x - (1/8)  0

While it might seem tempting to multiply the entire equation by the denominator to eliminate the fractions, doing so can lead to incorrect results. Instead, we will use the quadratic formula, which is a reliable method for solving such equations.

Using the Quadratic Formula

The quadratic formula is given by:

x  [-b ± √(b2 - 4ac)] / (2a)

Here, (a 1), (b 1/4), and (c -1/8). Plugging these values into the formula will give us the solutions for (x).

Step-by-Step Solution

Identify the coefficients: a 1 b 1/4 c -1/8 Calculate the discriminant: Δ b2 - 4ac Δ (1/4)2 - 4(1)(-1/8) Δ 1/16 1/2 Δ 1/16 8/16 Δ 9/16 Apply the quadratic formula: x [-b ± √Δ] / (2a) x [-1/4 ± √(9/16)] / (2 * 1) x [-1/4 ± 3/4] / 2 Calculate the two solutions: x1 (2/4) / 2 1/4 x2 (-4/4) / 2 -1/2

Thus, the solutions to the equation (x^2 (1/4)x - (1/8) 0) are x 1/4 and x -1/2.

Why Can't We Just Multiply by the Denominator?

Let's explore why simply multiplying the entire equation by the denominator to eliminate fractions doesn't work. Consider the example equation again:

x2   (1/4)x - (1/8)  0

If we multiply every term by 8 (the least common denominator of 1/4 and 1/8), we get:

8x2   2x - 1  0

Now, we would need to apply the quadratic formula to this new equation:

x  [-2 ± √(22 - 4(8)(-1))] / (2 * 8)

This approach can lead to confusing and incorrect results if you're not careful. Therefore, it is best to use the quadratic formula or other algebraic techniques to ensure accuracy.

Conclusion

Solving quadratic equations with fractions can be challenging, but understanding the quadratic formula and the importance of not simply multiplying by the denominator can help simplify the process. Always ensure you use the correct methods to avoid errors.

Additional Resources

For further reading and practice, you may want to explore resources on algebraic equations and the quadratic formula. Practice problems and interactive tools can also be found online to help solidify your understanding.