Question

Express x(x-3)(x-6)(x-9)+81 as perfect square.

Answer 1

SOLUTION

TO DETERMINE

Express x(x-3)(x-6)(x-9)+81 as perfect square.

EVALUATION

Here the given expression is

$\sf{x(x - 3)(x - 6)(x - 9) + 81}$

We simplify it as below

$\sf{x(x - 3)(x - 6)(x - 9) + 81}$

$\sf{ = x(x - 9)(x - 3)(x - 6) + 81}$

$\sf{ = ( {x}^{2} - 9x )( {x}^{2} - 9x + 18 ) + 81}$

Let y = x² - 9x

Then given expression becomes

$\sf{ = ( {x}^{2} - 9x )( {x}^{2} - 9x + 18 ) + 81}$

$\sf{ = y( y + 18 ) + 81}$

$\sf{ = {y}^{2} + 18y + 81}$

$\sf{ = {y}^{2} + 2 \times y \times 9+ {9}^{2} }$

$\sf{ = {(y + 9)}^{2} }$

$\sf{ = {( {x}^{2} - 9x + 9)}^{2} }$

Which is the required expression

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