Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation x^2+4*x+4=0
-
Equation (x^2-4)^2+(x^2-3x-10)^2=0
-
Equation -x/3+x+7=3
-
Equation -20*(x-13)=-220
- Express {x} in terms of y where:
- -14*x+4*y=-16
- -13*x-10*y=5
- 9*x-1*y=17
- -12*x-8*y=13
- Factor polynomial:
- z^2+z+1
- Derivative of:
-
z^2+z+1
-
Identical expressions
- z^ two +z+ one = zero
- z squared plus z plus 1 equally 0
- z to the power of two plus z plus one equally zero
- z2+z+1=0
- z²+z+1=0
- z to the power of 2+z+1=0
- z^2+z+1=O
-
Similar expressions
- z^2+z-1=0
- z^2-z+1=0
z^2+z+1=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$z_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$z_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = 1$$
$$c = 1$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$z_{1} = - \frac{1}{2} + \frac{\sqrt{3} i}{2}$$
$$z_{2} = - \frac{1}{2} - \frac{\sqrt{3} i}{2}$$
a*z^2 + b*z + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$z_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$z_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = 1$$
$$c = 1$$
, then
D = b^2 - 4 * a * c =
(1)^2 - 4 * (1) * (1) = -3
Because D<0, then the equation
has no real roots,
but complex roots is exists.
z1 = (-b + sqrt(D)) / (2*a)
z2 = (-b - sqrt(D)) / (2*a)
or
$$z_{1} = - \frac{1}{2} + \frac{\sqrt{3} i}{2}$$
$$z_{2} = - \frac{1}{2} - \frac{\sqrt{3} i}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p z + q + z^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 1$$
$$q = \frac{c}{a}$$
$$q = 1$$
Vieta Formulas
$$z_{1} + z_{2} = - p$$
$$z_{1} z_{2} = q$$
$$z_{1} + z_{2} = -1$$
$$z_{1} z_{2} = 1$$
$$p z + q + z^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 1$$
$$q = \frac{c}{a}$$
$$q = 1$$
Vieta Formulas
$$z_{1} + z_{2} = - p$$
$$z_{1} z_{2} = q$$
$$z_{1} + z_{2} = -1$$
$$z_{1} z_{2} = 1$$
Rapid solution
[src]
___
1 I*\/ 3
z1 = - - - -------
2 2
$$z_{1} = - \frac{1}{2} - \frac{\sqrt{3} i}{2}$$
___
1 I*\/ 3
z2 = - - + -------
2 2
$$z_{2} = - \frac{1}{2} + \frac{\sqrt{3} i}{2}$$
z2 = -1/2 + sqrt(3)*i/2
Sum and product of roots
[src]
sum
___ ___ 1 I*\/ 3 1 I*\/ 3 - - - ------- + - - + ------- 2 2 2 2
$$\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)$$
=
-1
$$-1$$
product
/ ___\ / ___\ | 1 I*\/ 3 | | 1 I*\/ 3 | |- - - -------|*|- - + -------| \ 2 2 / \ 2 2 /
$$\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)$$
=
1
$$1$$
1
Numerical answer
[src]
z1 = -0.5 - 0.866025403784439*i
z2 = -0.5 + 0.866025403784439*i
z2 = -0.5 + 0.866025403784439*i
The graph