Problem 333 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Solve the equation \(\frac{x^{2}+1}{a^{2} x-2 a}-\frac{1}{2-a x}=\frac{x}{a}\).

Step-by-Step Solution

Answer

The solutions are \(x = \frac{1}{a}\) and \(x = -1\)

Step 1: Multiply Through by Denominator

First multiply every term in the equation by \(a^{2} x - 2a\) to remove complex fractions. This will modify our equation into \(x^{2} + 1 - (a^{2} x - 2a) = ax^{2} - 2ax\). You would then need to simplify further.

Step 2: Re-arrange and Simplify

Combine like terms together, re-arrange the equation and simplify. This leads to the equation as \(x^{2} - ax^2 + a^{2} x - 1 = 0\), which can be sorted into the quadratic form \(ax^{2} - x^{2} + a^{2}x - 1 = 0\).

Step 3: Factorise and Solve

Finally, factorise the equation. It helps to recognize that the equation is a difference of two squares so it can be factorised as \((a x -1)(x + 1) = 0 \). Solve for \(x\) to get \(x = \frac{1}{a} or x = -1\)

Key Concepts

Equation Solving

Equation solving is the process of finding the values of variables that make an equation true.
Solving equations often involves several steps to isolate the variable and find its value.
To start with an equation like \( \frac{x^{2}+1}{a^{2} x-2 a}-\frac{1}{2-a x}=\frac{x}{a} \), the first step is to eliminate any fractions by multiplying through by a common denominator.
This simplifies the calculation and makes it easier to manipulate the equation.

• Identify a common denominator: The common denominator in our case is \( a^{2}x - 2a \).
• Multiply each term in the equation by this denominator to remove the fractions.

Strategically re-arranging terms such as moving all terms involving \( x \) to one side is crucial.
By simplifying the equation step-by-step, you will move closer to the solution.

Quadratic Equations

A quadratic equation is a specific type of polynomial equation that takes the form \( ax^2 + bx + c = 0 \).
The highest degree of the variable is 2, indicating that it will generally have two solutions.
In our problem, after simplification, we reach a form resembling a quadratic equation.

• Recognize the quadratic structure: Converting it to a standard form clarifies what to solve for.
• Apply techniques like completing the square or using the quadratic formula only if factorization is not evident.

Quadratic equations have distinct characteristics that make factorization an ideal method, particularly when coefficients are straightforward.
Our example, \( ax^{2} - x^{2} + a^{2}x - 1 = 0 \), demonstrates a form that lends itself well to factorization.

Factorization

Factorization involves rewriting an equation as a product of its factors.
This is particularly powerful with quadratic equations, where it can swiftly find solutions.
The equation \( (a x -1)(x + 1) = 0 \) is a factorized form indicating potential solutions within each bracket.

• Set each factor equal to zero: \( ax - 1 = 0 \) and \( x + 1 = 0 \) provide solutions for \( x \).
• This reveals \( x = \frac{1}{a} \) or \( x = -1 \) as solutions to the original quadratic equation.

By breaking down an equation into simpler multiplicative components, you directly find the equation's roots.
Factorization not only offers solutions but also confirms that these are indeed valid by substituting back into the original equation. Ultimately, mastery of factorization techniques enhances one's problem-solving arsenal effectively.