The determinate of the following 3x3 matrix(adsbygoogle = window.adsbygoogle || []).push({});

1-y, 2 , 3

2 , 4-y, 5

3 , 5 , 7-y

gives a cubic that simplifies to,

y^3 - 12*y^2 + y + 1 = 0.

Now, apparently the teacher picked random numbers for the original matrix, making the problem delve into other realms of mathematics. It isn't solvable by factoring methods, but using Cardano's method...

Letting,

y = x +4,

the cubic becomes,

(x + 4)^3 - 12*(x + 4)^2 + (x + 4) + 1 = 0.

This simplifies to,

x^3 - 47*x -123 = 0,

which is in the form

x^3 + b*x + c = 0.

To solve a cubic like this, b and c are just plugged into the following formula,

x = ( -(c/2) + ((c/2)^1/2 + (b/3)^1/3)^1/2)^1/3 + (-(c/2) - ((c/2)^1/2 + (b/3)^1/3)^1/2)^1/3,

which for my cubic comes out...

x = ((123/2) + (-6809/108)^1/2)^1/3 + ((123/2) - (-6809/108)^1/2)^1/3.

This is a real root, but I have no idea how to simplify it, ridding it of the complex parts.

Sorry to show all the lead up, it really wasn't needed for the simplification, but since the rules say to show what you have done... I did.

Thanks in advance for any help.

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# Homework Help: Solving a cubic (or rather simplifying a real root in complex form)

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