What must be added to each of the following expressions to make it a whole square?

Solution:

(i) Let us consider the following expression:

$4 x^{2}-12 x+7$

The above expression can be written as:

$4 x^{2}-12 x+7=(2 x)^{2}-2 \times 2 x \times 3+7$

It is evident that if 2x is considered as the first term and 3 is considered as the second term, 2 is required to be added to the above expression to make it a perfect square. Therefore, 7 must become 9.

Therefore, adding and subtracting 2 in the above expression, we get:

$\left(4 x^{2}-12 x+7\right)+2-2=\left\{(2 x)^{2}-2 \times 2 x \times 3+7\right\}+2-2=\left\{(2 x)^{2}-2 \times 2 x \times 3+9\right\}-2=(2 x+3)^{2}-2$

Thus, the answer is 2.

(ii) Let's consider the following expression:

$4 x^{2}-20 x+20$

The above expression can be written as:

$4 x^{2}-20 x+20=(2 x)^{2}-2 \times 2 x \times 5+20$

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:

$\left(4 x^{2}-20 x+20+5\right)-5=\left\{(2 x)^{2}-2 \times 2 x \times 5+20\right\}+5-5=\left\{(2 x)^{2}-2 \times 2 x \times 5+25\right\}-5=(2 x+5)^{2}-5$

Thus, the answer is 5.