# What must be added to 81x^2 - 72x + 9 to make it a perfect square?

Mar 31, 2016

A constant solution would be $7$

Another solution is $18 x$

#### Explanation:

$81 {x}^{2} - 72 x + 9 + 7 = 81 {x}^{2} - 72 x + 16 = {\left(9 x - 4\right)}^{2}$

$81 {x}^{2} - 72 x + 9 + 18 x = 81 {x}^{2} - 54 x + 9 = {\left(9 x - 3\right)}^{2}$

In general,

${\left(a - b\right)}^{2} = {a}^{2} - 2 a b + {b}^{2}$

In the above solutions, we choose to change the constant term to match the $- 2 a b$ term or the $- 2 a b$ term to match the ${b}^{2}$ term.