solve the equation 3x^3-17x^2-8x+12=0 given that the product of two of the roots is 4

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should use Vieta's relations, that link the three roots of equation to coefficients of equation, such that:

x_1 + x_2 + x_3 = 17/3

x_1x_2 + x_1x_3 + x_2x_3 = -8/3

x_1x_2x_3 = -12/3

Notice that the problem provides the information that the product of two roots is 4,hence, you should consider `x_1x_2 = 4`  such that:

`x_1x_2x_3 = -4 => 4x_3 = -4 => x_3 = -1`

Substituting -1 for `x_3`  in the first relation yields:

`x_1 + x_2 - 1 = 17/3 => x_1 + x_2 = 17/3 + 1 => x_1 + x_2 = 20/3`

You should use Lagrange's resolvents such that:

`x^2 - (x_1+x_2)*x + x_1x_2 = 0`

`x^2 - 20/3x + 4 = 0 => 3x^2 - 20x + 12 = 0`

Using quadratic formula yields:

`x_(1,2) = (20+-sqrt(400 - 144))/6`

`x_(1,2) = (20+-sqrt256)/6 => x_(1,2) = (20+-16)/6`

`x_1 = 6 ; x_2 = 2/3`

Hence, evaluating the soutions to the given equation, under the given conditions, yields `x_1 = 6 ; x_2 = 2/3 ; x_3 = -1` .

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