Question

Prove that x² − 6x + 9 is a perfect square for all natural numbers x. What can you say about its square root?

Answer 1

To show that the expression x² − 6x + 9 is a perfect square, we need to show that it can be written as the square of a linear expression.

⇒ x² − 6x + 9 = x² − 2 × 3 × x + 9

= (x − 3)² {(x − y)² = x² + y² − 2xy}

∴ x² − 6x + 9 = (x − 3)²

Hence, x² − 6x + 9 is a perfect square.

Square root of x² − 6x + 9 =

=

= x − 3