Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation -3x-9=2x
-
Equation x^2+6=0
-
Equation (3x-8)^2-(4x-6)^2+(5x-2)(5x+2)=96
-
Equation 2*x-2=0
- Express {x} in terms of y where:
- -19*x+1*y=0
- 9*x+17*y=13
- -1*x+4*y=-10
- -18*x+12*y=-8
- Derivative of:
- x^3+3x^2+2x
- Graphing y =:
- x^3+3x^2+2x
-
Identical expressions
- x^ three +3x^ two +2x= two hundred and ten
- x cubed plus 3x squared plus 2x equally 210
- x to the power of three plus 3x to the power of two plus 2x equally two hundred and ten
- x3+3x2+2x=210
- x³+3x²+2x=210
- x to the power of 3+3x to the power of 2+2x=210
-
Similar expressions
- (1-x^2*9.79*1.12*10^(-5))*(1-x^2*14.69*1.95*10^(-7))-x^4*9.79*14.69*(5.59*10^(-6))^2=0
- 2*1000*(0.28-x)=800*x
- x^3-3x^2+2x=210
- x^3+3x^2-2x=210
- 2^(10x^2-8x-23)+2^(5x^2-4x-12)-3=0
x^3+3x^2+2x=210 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$2 x + \left(x^{3} + 3 x^{2}\right) = 210$$
transform
$$\left(2 x + \left(\left(3 x^{2} + \left(x^{3} - 125\right)\right) - 75\right)\right) - 10 = 0$$
or
$$\left(2 x + \left(\left(3 x^{2} + \left(x^{3} - 5^{3}\right)\right) - 3 \cdot 5^{2}\right)\right) + \left(-2\right) 5 = 0$$
$$2 \left(x - 5\right) + \left(3 \left(x^{2} - 5^{2}\right) + \left(x^{3} - 5^{3}\right)\right) = 0$$
$$2 \left(x - 5\right) + \left(\left(x - 5\right) \left(\left(x^{2} + 5 x\right) + 5^{2}\right) + 3 \left(x - 5\right) \left(x + 5\right)\right) = 0$$
Take common factor -5 + x from the equation
we get:
$$\left(x - 5\right) \left(\left(3 \left(x + 5\right) + \left(\left(x^{2} + 5 x\right) + 5^{2}\right)\right) + 2\right) = 0$$
or
$$\left(x - 5\right) \left(x^{2} + 8 x + 42\right) = 0$$
then:
$$x_{1} = 5$$
and also
we get the equation
$$x^{2} + 8 x + 42 = 0$$
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
$$a = 1$$
$$b = 8$$
$$c = 42$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{2} = -4 + \sqrt{26} i$$
$$x_{3} = -4 - \sqrt{26} i$$
The final answer for x^3 + 3*x^2 + 2*x - 210 = 0:
$$x_{1} = 5$$
$$x_{2} = -4 + \sqrt{26} i$$
$$x_{3} = -4 - \sqrt{26} i$$
$$2 x + \left(x^{3} + 3 x^{2}\right) = 210$$
transform
$$\left(2 x + \left(\left(3 x^{2} + \left(x^{3} - 125\right)\right) - 75\right)\right) - 10 = 0$$
or
$$\left(2 x + \left(\left(3 x^{2} + \left(x^{3} - 5^{3}\right)\right) - 3 \cdot 5^{2}\right)\right) + \left(-2\right) 5 = 0$$
$$2 \left(x - 5\right) + \left(3 \left(x^{2} - 5^{2}\right) + \left(x^{3} - 5^{3}\right)\right) = 0$$
$$2 \left(x - 5\right) + \left(\left(x - 5\right) \left(\left(x^{2} + 5 x\right) + 5^{2}\right) + 3 \left(x - 5\right) \left(x + 5\right)\right) = 0$$
Take common factor -5 + x from the equation
we get:
$$\left(x - 5\right) \left(\left(3 \left(x + 5\right) + \left(\left(x^{2} + 5 x\right) + 5^{2}\right)\right) + 2\right) = 0$$
or
$$\left(x - 5\right) \left(x^{2} + 8 x + 42\right) = 0$$
then:
$$x_{1} = 5$$
and also
we get the equation
$$x^{2} + 8 x + 42 = 0$$
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
$$a = 1$$
$$b = 8$$
$$c = 42$$
, then
D = b^2 - 4 * a * c =
(8)^2 - 4 * (1) * (42) = -104
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
$$x_{2} = -4 + \sqrt{26} i$$
$$x_{3} = -4 - \sqrt{26} i$$
The final answer for x^3 + 3*x^2 + 2*x - 210 = 0:
$$x_{1} = 5$$
$$x_{2} = -4 + \sqrt{26} i$$
$$x_{3} = -4 - \sqrt{26} i$$
Vieta's Theorem
it is reduced cubic equation
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 3$$
$$q = \frac{c}{a}$$
$$q = 2$$
$$v = \frac{d}{a}$$
$$v = -210$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -3$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 2$$
$$x_{1} x_{2} x_{3} = -210$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 3$$
$$q = \frac{c}{a}$$
$$q = 2$$
$$v = \frac{d}{a}$$
$$v = -210$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -3$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 2$$
$$x_{1} x_{2} x_{3} = -210$$
Rapid solution
[src]
x1 = 5
$$x_{1} = 5$$
____ x2 = -4 - I*\/ 26
$$x_{2} = -4 - \sqrt{26} i$$
____ x3 = -4 + I*\/ 26
$$x_{3} = -4 + \sqrt{26} i$$
x3 = -4 + sqrt(26)*i
Sum and product of roots
[src]
sum
____ ____ 5 + -4 - I*\/ 26 + -4 + I*\/ 26
$$\left(5 + \left(-4 - \sqrt{26} i\right)\right) + \left(-4 + \sqrt{26} i\right)$$
=
-3
$$-3$$
product
/ ____\ / ____\ 5*\-4 - I*\/ 26 /*\-4 + I*\/ 26 /
$$5 \left(-4 - \sqrt{26} i\right) \left(-4 + \sqrt{26} i\right)$$
=
210
$$210$$
210
Numerical answer
[src]
x1 = 5.0
x2 = -4.0 + 5.09901951359278*i
x3 = -4.0 - 5.09901951359278*i
x3 = -4.0 - 5.09901951359278*i