Quadratic equations

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

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