Question

Match the expressions on the left with the number to be added to it to make the resultant a perfect square.

• A. 81
• B. 4
• C. 121

Answer 1

The identity $(a+b{)}^2=(a^2+2ab+b^2)$ represents a perfect square.

Consider the expression $x^2+18x.$
If we halve the coefficient of $x$ and add its square to this expression, we get
$x^2+18x+81.$

But,
$x^2+18x+81=x^2+2\left(x\right)\left(9\right)+(9{)}^2$
$=(x+9{)}^2,$ a perfect square.

Thus, we have to add 81 to the expression $(x^2+18x)$ so that the resultant is a perfect square.

Similarly, 4 must be added to $x^2+4x$ so that the sum is a perfect square as shown below:
$x^2+4x+4=x^2+2\left(x\right)\left(2\right)+(2{)}^2$
$=(x+2{)}^2$

And, 121 must be added to $x^2+22x$ to make the resultant a perfect square.
$x^2+22x+121=(x+11{)}^2$