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Equation x^3+9*x^2+27*x+27=0
- Express {x} in terms of y where:
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- Factor polynomial:
- x^3+9*x^2+27*x+27
-
Identical expressions
- x^ three + nine *x^ two + twenty-seven *x+ twenty-seven = zero
- x cubed plus 9 multiply by x squared plus 27 multiply by x plus 27 equally 0
- x to the power of three plus nine multiply by x to the power of two plus twenty minus seven multiply by x plus twenty minus seven equally zero
- x3+9*x2+27*x+27=0
- x³+9*x²+27*x+27=0
- x to the power of 3+9*x to the power of 2+27*x+27=0
- x^3+9x^2+27x+27=0
- x3+9x2+27x+27=0
- x^3+9*x^2+27*x+27=O
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Similar expressions
- x^3+9*x^2+27*x-27=0
- x^3-9*x^2+27*x+27=0
- x^3+9*x^2-27*x+27=0
x^3+9*x^2+27*x+27=0 equation
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The solution
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3 2 x + 9*x + 27*x + 27 = 0
$$\left(27 x + \left(x^{3} + 9 x^{2}\right)\right) + 27 = 0$$
Detail solution
Given the equation:
$$\left(27 x + \left(x^{3} + 9 x^{2}\right)\right) + 27 = 0$$
transform
$$\left(27 x + \left(\left(9 x^{2} + \left(x^{3} + 27\right)\right) - 81\right)\right) + 81 = 0$$
or
$$\left(27 x + \left(\left(9 x^{2} + \left(x^{3} - \left(-3\right)^{3}\right)\right) - 9 \left(-3\right)^{2}\right)\right) - -81 = 0$$
$$27 \left(x + 3\right) + \left(9 \left(x^{2} - \left(-3\right)^{2}\right) + \left(x^{3} - \left(-3\right)^{3}\right)\right) = 0$$
$$27 \left(x + 3\right) + \left(\left(x - 3\right) 9 \left(x + 3\right) + \left(x + 3\right) \left(\left(x^{2} - 3 x\right) + \left(-3\right)^{2}\right)\right) = 0$$
Take common factor 3 + x from the equation
we get:
$$\left(x + 3\right) \left(\left(9 \left(x - 3\right) + \left(\left(x^{2} - 3 x\right) + \left(-3\right)^{2}\right)\right) + 27\right) = 0$$
or
$$\left(x + 3\right) \left(x^{2} + 6 x + 9\right) = 0$$
then:
$$x_{1} = -3$$
and also
we get the equation
$$x^{2} + 6 x + 9 = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{2} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{3} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = 6$$
$$c = 9$$
, then
Because D = 0, then the equation has one root.
$$x_{2} = -3$$
The final answer for x^3 + 9*x^2 + 27*x + 27 = 0:
$$x_{1} = -3$$
$$x_{2} = -3$$
$$\left(27 x + \left(x^{3} + 9 x^{2}\right)\right) + 27 = 0$$
transform
$$\left(27 x + \left(\left(9 x^{2} + \left(x^{3} + 27\right)\right) - 81\right)\right) + 81 = 0$$
or
$$\left(27 x + \left(\left(9 x^{2} + \left(x^{3} - \left(-3\right)^{3}\right)\right) - 9 \left(-3\right)^{2}\right)\right) - -81 = 0$$
$$27 \left(x + 3\right) + \left(9 \left(x^{2} - \left(-3\right)^{2}\right) + \left(x^{3} - \left(-3\right)^{3}\right)\right) = 0$$
$$27 \left(x + 3\right) + \left(\left(x - 3\right) 9 \left(x + 3\right) + \left(x + 3\right) \left(\left(x^{2} - 3 x\right) + \left(-3\right)^{2}\right)\right) = 0$$
Take common factor 3 + x from the equation
we get:
$$\left(x + 3\right) \left(\left(9 \left(x - 3\right) + \left(\left(x^{2} - 3 x\right) + \left(-3\right)^{2}\right)\right) + 27\right) = 0$$
or
$$\left(x + 3\right) \left(x^{2} + 6 x + 9\right) = 0$$
then:
$$x_{1} = -3$$
and also
we get the equation
$$x^{2} + 6 x + 9 = 0$$
This equation is of the form
a*x^2 + b*x + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{2} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{3} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = 6$$
$$c = 9$$
, then
D = b^2 - 4 * a * c =
(6)^2 - 4 * (1) * (9) = 0
Because D = 0, then the equation has one root.
x = -b/2a = -6/2/(1)
$$x_{2} = -3$$
The final answer for x^3 + 9*x^2 + 27*x + 27 = 0:
$$x_{1} = -3$$
$$x_{2} = -3$$
Vieta's Theorem
it is reduced cubic equation
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 9$$
$$q = \frac{c}{a}$$
$$q = 27$$
$$v = \frac{d}{a}$$
$$v = 27$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -9$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 27$$
$$x_{1} x_{2} x_{3} = 27$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 9$$
$$q = \frac{c}{a}$$
$$q = 27$$
$$v = \frac{d}{a}$$
$$v = 27$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -9$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 27$$
$$x_{1} x_{2} x_{3} = 27$$
The graph