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

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Answer 1 · 2016-03-31T22:01:04

A constant solution would be $7$

Another solution is $18x$

$81x^2-72x+9+7 = 81x^2-72x+16 = (9x-4)^2$

$81x^2-72x+9+18x = 81x^2-54x+9 = (9x-3)^2$

In general,

$(a-b)^2 = a^2-2ab+b^2$

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

What must be added to 81x^2 - 72x + 981x^2 - 72x + 9 to make it a perfect square?