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Posted by Mark on November 07, 2002 at 11:44:10:

I have an algebra problem involving the simplification of radicals, which I have been unable to solve satisfactorily.

The objective is to simplify and collect: SQRT[3x^2 - 18x + 27] - SQRT[27(x^2 + 2x + 1)].

My solution:

SQRT[3x^2 - 18x + 27] - SQRT[27(x^2 + 2x + 1)] = SQRT[(3)(x^2 - 6x + 9)] - SQRT[(9)(3)(x^2 + 2x + 1)]
= SQRT[(3)(x - 3)(x - 3)] - SQRT[(3^2)(3)(x + 1)(x + 1)]
= SQRT[(3)(x - 3)^2] - SQRT[(3^2)(3)(x + 1)^2]
= (x - 3)SQRT3 - [(3)(x + 1)]SQRT3
= (x - 3)SQRT3 - (3x + 3)SQRT3
= [(x - 3) - (3x + 3)]SQRT3
= (x - 3 - 3x - 3)SQRT3
= (-2x - 6)SQRT3

I checked by substituting 2 for x:
SQRT[3(2)^2 - 18(2) + 27] - SQRT[27((2)^2 + 2(2) + 1)] = (-2(2) - 6)SQRT3
SQRT[12 - 36 + 27] - SQRT[27(4 + 4 + 1)] = (-4 - 6)SQRT3
SQRT3 - SQRT[27(9)] = -10SQRT3
SQRT3 - SQRT[(3^2)(3)(3^2)] = -10SQRT3
SQRT3 - (3)(3)SQRT3 = -10SQRT3
SQRT3 - (9)SQRT3 = -10SQRT3
(1 - 9)SQRT3 = -10SQRT3
-8SQRT3 = -10SQRT3

I have been over this several times, and I am out of ideas. Where is my error?

Thanks for the help.

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