Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation (2*x-6)^2-4*x^2=0
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Equation x^3+9*x^2+27*x+27=0
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Equation x^2-7*x+6=0
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Equation x^2+3x+1=0
- Express {x} in terms of y where:
- -20*x+6*y=-12
- -11*x+16*y=-5
- -12*x-9*y=-3
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Identical expressions
- x^ three -6x- forty = zero
- x cubed minus 6x minus 40 equally 0
- x to the power of three minus 6x minus forty equally zero
- x3-6x-40=0
- x³-6x-40=0
- x to the power of 3-6x-40=0
- x^3-6x-40=O
-
Similar expressions
- x^3-6x+40=0
- x^3+6x-40=0
x^3-6x-40=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$\left(x^{3} - 6 x\right) - 40 = 0$$
transform
$$\left(- 6 x + \left(x^{3} - 64\right)\right) + 24 = 0$$
or
$$\left(- 6 x + \left(x^{3} - 4^{3}\right)\right) + 4 \cdot 6 = 0$$
$$- 6 \left(x - 4\right) + \left(x^{3} - 4^{3}\right) = 0$$
$$\left(x - 4\right) \left(\left(x^{2} + 4 x\right) + 4^{2}\right) - 6 \left(x - 4\right) = 0$$
Take common factor -4 + x from the equation
we get:
$$\left(x - 4\right) \left(\left(\left(x^{2} + 4 x\right) + 4^{2}\right) - 6\right) = 0$$
or
$$\left(x - 4\right) \left(x^{2} + 4 x + 10\right) = 0$$
then:
$$x_{1} = 4$$
and also
we get the equation
$$x^{2} + 4 x + 10 = 0$$
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
$$a = 1$$
$$b = 4$$
$$c = 10$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{2} = -2 + \sqrt{6} i$$
$$x_{3} = -2 - \sqrt{6} i$$
The final answer for x^3 - 6*x - 40 = 0:
$$x_{1} = 4$$
$$x_{2} = -2 + \sqrt{6} i$$
$$x_{3} = -2 - \sqrt{6} i$$
$$\left(x^{3} - 6 x\right) - 40 = 0$$
transform
$$\left(- 6 x + \left(x^{3} - 64\right)\right) + 24 = 0$$
or
$$\left(- 6 x + \left(x^{3} - 4^{3}\right)\right) + 4 \cdot 6 = 0$$
$$- 6 \left(x - 4\right) + \left(x^{3} - 4^{3}\right) = 0$$
$$\left(x - 4\right) \left(\left(x^{2} + 4 x\right) + 4^{2}\right) - 6 \left(x - 4\right) = 0$$
Take common factor -4 + x from the equation
we get:
$$\left(x - 4\right) \left(\left(\left(x^{2} + 4 x\right) + 4^{2}\right) - 6\right) = 0$$
or
$$\left(x - 4\right) \left(x^{2} + 4 x + 10\right) = 0$$
then:
$$x_{1} = 4$$
and also
we get the equation
$$x^{2} + 4 x + 10 = 0$$
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
$$a = 1$$
$$b = 4$$
$$c = 10$$
, then
D = b^2 - 4 * a * c =
(4)^2 - 4 * (1) * (10) = -24
Because D<0, then the equation
has no real roots,
but complex roots is exists.
x2 = (-b + sqrt(D)) / (2*a)
x3 = (-b - sqrt(D)) / (2*a)
or
$$x_{2} = -2 + \sqrt{6} i$$
$$x_{3} = -2 - \sqrt{6} i$$
The final answer for x^3 - 6*x - 40 = 0:
$$x_{1} = 4$$
$$x_{2} = -2 + \sqrt{6} i$$
$$x_{3} = -2 - \sqrt{6} i$$
Vieta's Theorem
it is reduced cubic equation
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -6$$
$$v = \frac{d}{a}$$
$$v = -40$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -6$$
$$x_{1} x_{2} x_{3} = -40$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -6$$
$$v = \frac{d}{a}$$
$$v = -40$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -6$$
$$x_{1} x_{2} x_{3} = -40$$
Rapid solution
[src]
x1 = 4
$$x_{1} = 4$$
___ x2 = -2 - I*\/ 6
$$x_{2} = -2 - \sqrt{6} i$$
___ x3 = -2 + I*\/ 6
$$x_{3} = -2 + \sqrt{6} i$$
x3 = -2 + sqrt(6)*i
Sum and product of roots
[src]
sum
___ ___ 4 + -2 - I*\/ 6 + -2 + I*\/ 6
$$\left(4 + \left(-2 - \sqrt{6} i\right)\right) + \left(-2 + \sqrt{6} i\right)$$
=
0
$$0$$
product
/ ___\ / ___\ 4*\-2 - I*\/ 6 /*\-2 + I*\/ 6 /
$$4 \left(-2 - \sqrt{6} i\right) \left(-2 + \sqrt{6} i\right)$$
=
40
$$40$$
40
Numerical answer
[src]
x1 = -2.0 - 2.44948974278318*i
x2 = -2.0 + 2.44948974278318*i
x3 = 4.0
x3 = 4.0