Given the law of composition x*y = 2xy - 6x - 6y + 21 solve the equation x*x*x = 3

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neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

Posted on

x*y =2xy-6x-6y+21

To find x*x*x.

Under the composition we find x*x and then x*x*x.

x*x = 2x^2-6x-6x+21 = 2x^2 -12x+21.

x*x*x = 2(2x^2-12x+21)x-6(2x^2-12x+21)-6x+21

x*x*x = 4x^3-24x^2+42x -12x^2+72x-126-6x+21

x*x*x = 4x^3 -36x^2 +(42+72-6)x -105

x*x*x = 4x^3 -36x^2 +108x-105

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll re-write the law of composition as:

 2xy - 6x - 6y + 21 = 2(x-3)(y-3) + 3

Now, we'll apply the law of composition for x*x:

x*x = 2(x-3)(x-3) + 3

x*x = 2(x-3)^2 + 3

Now we'll solve x*x*x:

[2(x-3)^2 + 3]*x = 2[2(x-3)^2 + 3 - 3](x-3) + 3

We'll eliminate like terms:

4(x-3)^3 + 3 = 3

We'll eliminate like terms:

4(x-3)^3 = 0

We'll divide by 4:

(x-3)^3 = 0

The solutions of the equation are: x1 = x2 = x3  = 3.

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