Question

What must be added to the following expression to make it a whole square? 

4x² − 20x + 20

Answer 1

Let's consider the following expression: \[4 x^2 - 20x + 20\]

The above expression can be written as: \[4 x^2 - 20x + 20 = \left( 2x \right)^2 - 2 \times 2x \times 5 + 20\]

It is evident that if 2x is considered as the first term and 5 is considered as the second term, 5 is required to be added to the above expression to make it a perfect square. Therefore, number 20 must become 25.

Therefore, adding and subtracting 5 in the above expression, we get:

\[\left( 4 x^2 - 20x + 20 + 5 \right) - 5 = \left\{ \left( 2x \right)^2 - 2 \times 2x \times 5 + 20 \right\} + 5 - 5 = \left\{ \left( 2x \right)^2 - 2 \times 2x \times 5 + 25 \right\} - 5 = \left( 2x + 5 \right)^2 - 5\] Thus, the answer is 5.