# Solve the following equation in the real number system. Please show all of your work. `x^3-11x^2+24x-12=0` ``not calculus

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`x^3 -11x^2 +24x -12 = 0`

`` First we will rewrite the equation by rewriting -`11x^2 as -2x^2-9x^2` .

`x^3 - 2x^2 -9x^2 +24x -12 = 0`

`` Now we will group terms:

`(x^3 -2x^2) - (9x^2 -24x +12) = 0`

`` Now we will factor:

`x^2 (x-2) - (9x-6)(x-2)= 0`

`` Now we will factor (x-2).

`==gt (x-2) (x^2 -(9x-6)) = 0 `

`==gt (x-2)(x^2 -9x +6) = 0`

`` ==> `x_1 = 2`

`` ==> `x_2 = (9+sqrt(81-24))/2 = (9+sqrt57)/2`

`` ==> `x_3 = (9-sqrt57)/2`

`==> x = { 2, (9+sqrt57)/2 , (9-sqrt57)/2 }`

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