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Question

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

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Solution

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


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