Worked Out Example 1
If the following expression is a perfect square, find the value of a
A perfect square expression can be transformed into complete square and vice versa. The above expression is written in complete square form. Now, change it into a perfect square as it is demonstrated here
Suppose you let
Remember the m plus n squared is a perfect square. A perfect square is a contracted form of a complete square.
Expand the RHS to get
As you can see elements on the LHS equals elements on the RHS. For clarity purposes equate same terms on both sides.
Equate first terms
Equate second terms
Equate third terms
By observation, it is much easier to start solving from the second term because it is simpler. It comprises of first degree polynomial.
By dividing both sides by 2 the equations reduces to
Square both sides. It rewrites to
Which simplifies to
As for the RHS, this is what you need for substitution
From the above two equations, it becomes apparent that
Make a the subject
Express the RHS as a product of its prime factors i.e factorize the numerator and denominator to prepare the equation for simplification
Now simplify by cancelling
Your leaner equation is
Your final solution becomes