Solving Quadratic Equations with Rational Expressions: A Step-by-Step Guide
Quadratic equations can sometimes present themselves in complex forms, such as rational expressions. In this article, we will explore how to tackle equations of the form (frac{1}{x1} frac{1}{x-1} frac{1}{x2} frac{1}{x-2} 0). We'll break down the process step-by-step and explore the underlying mathematics.
Understanding the Given Equation
The given equation is:
[frac{1}{x-1} frac{1}{x-2} frac{1}{x1} frac{1}{x2} 0]Initially, it might seem challenging due to the presence of these rational expressions. However, by following a methodical approach, we can simplify and solve the equation.
Step-by-Step Solution
To solve this equation, we start by finding a common denominator for the left-hand side:
[frac{1}{x-1} frac{1}{x-2} frac{1}{x1} frac{1}{x2} 0]The common denominator is ( (x-1)(x-2)(x1)(x2) ). Multiplying both sides of the equation by this common denominator, we get:
[frac{(x-1)(x-2)(x1)(x2)}{(x-1)(x-2)(x1)(x2)} cdot left( frac{1}{x-1} frac{1}{x-2} frac{1}{x1} frac{1}{x2} right) 0 cdot (x-1)(x-2)(x1)(x2)]When we multiply through, we can simplify the equation as follows:
[(x-2)(x1)(x2) (x-1)(x1)(x2) (x-2)(x-1)(x2) (x-2)(x-1)(x1) 0]This simplifies to:
[frac{(x-2)(x1)(x2) (x-1)(x1)(x2) (x-2)(x-1)(x2) (x-2)(x-1)(x1)}{(x-1)(x-2)(x1)(x2)} 0]Since the denominator isn't zero (as (x eq 1, 2, -1, -2)), the numerator must be zero:
[(x-2)(x1)(x2) (x-1)(x1)(x2) (x-2)(x-1)(x2) (x-2)(x-1)(x1) 0]This equation is quite complex, and solving it directly can be intricate. Let's try another method to simplify it:
[frac{2x}{x^2-1} frac{2x}{x^2-4} 0]Multiplying both sides by ((x^2-1)(x^2-4)), we get:
[(2x)(x^2-4) (2x)(x^2-1) 0]Expanding the equation, we obtain:
[2x^3 - 8x 2x^3 - 2x 0]Combining like terms, we have:
[4x^3 - 1 0]This can be factored as:
[4x(x^2 - frac{5}{2}) 0]Setting each factor to zero:
[4x 0 quad text{or} quad x^2 - frac{5}{2} 0]Thus, the solutions are:
[x 0 quad text{or} quad x pm sqrt{frac{5}{2}}]Graphical Interpretation
The solutions to this equation can be visualized through the graph of the function. The function ( y frac{1}{x-1} frac{1}{x-2} frac{1}{x1} frac{1}{x2} ) has asymptotes at ( x 1, 2, -1, -2 ).
By analyzing the graph, we can observe that the function exhibits behavior similar to the graph of a quartic polynomial, indicating the presence of multiple roots. The roots of the equation (frac{1}{x-1} frac{1}{x-2} frac{1}{x1} frac{1}{x2} 0) are:
[boxed{x 0, pm sqrt{frac{5}{2}}}]Conclusion
In conclusion, solving the given equation through algebraic manipulation and simplification reveals the roots (x 0) and (x pm sqrt{frac{5}{2}}). Understanding the underlying principles of rational expressions and their manipulation is key to solving such complex equations.