Learn to Solve hard MCQ algebra problems fast
Many difficult MCQ algebra problems in major competitive tests such as SSC CGL can be solved easily and quickly if you use inventive principles and methods.
I will explain how by solving a carefully chosen example problem.
Chosen Example Algebra Problem for Lightning Quick Solution
The value of, $\displaystyle\frac{1}{a^2 +ab + b^2}- \displaystyle\frac{1}{a^2 - ab + b^2} +\displaystyle\frac{2ab}{a^4 + a^2b^2 + b^4}$ is,
- 2
- 1
- -1
- 0
Key Problem-Solving Techniques Used for the Quick Solution
Free Resource Use Principle
This principle urges you to use any available resource that is free, effective, and helpful in solving a problem without extra cost and in shortest time. Often, these resources are overlooked, but identifying and using them can significantly simplify your solution process.
In the example problem, the numeric choice values without variables are your free resources. These indicate: there must be a pattern of common factors between the denominators of the three given terms for combining the terms. Isn't it?
Problem Breakdown Technique
This is a frequently used, extended common sensed based technique of breaking up a complex problem into a number of simple parts and solve for each before combining the results.
In the example problem, I will consider combining two of the three terms first. Next I will combine the result of the first combining effort with the third term. Which of the two terms to combine first? The first two terms are best candidates as the third term is more complex in comparison.
This choice is well supported by another frequently used problem solving technique of solving a simpler problem.
The Technique of Solving a Simpler Problem
Combining the simpler first two terms is a part of the more complex three-term expression. It should be easier to combine the two. Follows common sense logic. The simpler problem I will solve first is: Simplify,
$\displaystyle\frac{1}{a^2 +ab + b^2}- \frac{1}{a^2 - ab + b^2}$
Pattern Recognition Technique
Pattern recognition and use underlies all problems in the world. It is so general. For example, a child learns to speak by identifying the patterns in the speech of the adults.
In the simpler problem, what is the pattern common between the denominators of the two terms? For combining the two terms, I remember the classic algebraic identity that must be used (by mentally rearranging the three terms in the two denominators),
$(x+y)(x-y) = x^2 - y^2$, where $x=(a^2 + b^2)$ and $y=ab$.
This key pattern is hidden and I rearranged the three terms in each denominator to uncover the key pattern.
The combined denominator simplifies to,
$(x^2 - y^2)=(a^2 + b^2)^2 - (ab)^2 = a^4 + a^2b^2 + b^4$.
And the numerator is simply,
$-2y=-2ab$.
The result of combining the first two terms,
$\displaystyle\frac{-2ab}{a^4 + a^2b^2 + b^4}$.
Solving the whole problem: Combine the result of the first simplified problem with the rest
The denominator is exactly same as the denominator of the third term. The two terms cancel out with answer as $0$.
To show this, I will now combine the result of the simplified problem with the third term (following the principles of problem breakdown technique),
$\displaystyle\frac{-2ab}{a^4 + a^2b^2 + b^4} + \displaystyle\frac{2ab}{a^4 + a^2b^2 + b^4} = 0$.
Answer: Option d: $0$.
Deductive reasoning binds the whole process of problem solving
At every stage, your deductive reasoning skill analyzes the present state and future prospects and helps you to select the right problem solving strategy and technique so that ultimately you achieve the most efficient solution in the shortest time.
If the powerful problem solving concepts are the flowers, deductive reasoning is the thread that connects the flowers into the garland of a beautiful solution!
End note
You may not be familiar with the problem solving techniques used, but the techniques are so natural that most of the fastest solvers of this problem would have surely used these techniques without being aware of the specific names or the details of the techniques.
Armed with awareness of how and where to use such problem solving techniques, any of you should solve such problem well within a minute, that too without using pen and paper. After all, you must be aware that for achieving high scores in the competitive exams like SSC CGL, you have to solve most of the problems mentally.
Are these special tricks? Absolutely NO.
Can this ability be attained in a very short time? We won't say, impossible. But yes, it is difficult and needs systematic and intelligent actual solving of complex problems under time pressure repeatedly.
We have enough problems solved using problem solving techniques. Use these free and valuable resources to become a high power problem solver
Free and valuable resources for solving SSC CGL algebra problems lightning fast
The list of Difficult algebra problem solving in a few steps quickly is available at, Quick algebra.
Go through the extended resource of powerful concepts and methods to solve difficult algebra problems easy and quick,