Mister Exam
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Similar expressions
- z^2-i=0
z^2+i=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
$$a = 1$$
$$b = 0$$
$$c = i$$
, then
The equation has two roots.
or
$$z_{1} = \sqrt{- i}$$
$$z_{2} = - \sqrt{- i}$$
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
$$a = 1$$
$$b = 0$$
$$c = i$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (1) * (i) = -4*i
The equation has two roots.
z1 = (-b + sqrt(D)) / (2*a)
z2 = (-b - sqrt(D)) / (2*a)
or
$$z_{1} = \sqrt{- i}$$
$$z_{2} = - \sqrt{- i}$$
Vieta's Theorem
it is reduced quadratic equation
$$p z + q + z^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = i$$
Vieta Formulas
$$z_{1} + z_{2} = - p$$
$$z_{1} z_{2} = q$$
$$z_{1} + z_{2} = 0$$
$$z_{1} z_{2} = i$$
$$p z + q + z^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = i$$
Vieta Formulas
$$z_{1} + z_{2} = - p$$
$$z_{1} z_{2} = q$$
$$z_{1} + z_{2} = 0$$
$$z_{1} z_{2} = i$$
The graph
Rapid solution
[src]
___ ___
\/ 2 I*\/ 2
z1 = ----- - -------
2 2
$$z_{1} = \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}$$
___ ___
\/ 2 I*\/ 2
z2 = - ----- + -------
2 2
$$z_{2} = - \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}$$
z2 = -sqrt(2)/2 + sqrt(2)*i/2
Sum and product of roots
[src]
sum
___ ___ ___ ___ \/ 2 I*\/ 2 \/ 2 I*\/ 2 ----- - ------- + - ----- + ------- 2 2 2 2
$$\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}\right) + \left(- \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}\right)$$
=
0
$$0$$
product
/ ___ ___\ / ___ ___\ |\/ 2 I*\/ 2 | | \/ 2 I*\/ 2 | |----- - -------|*|- ----- + -------| \ 2 2 / \ 2 2 /
$$\left(- \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}\right) \left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}\right)$$
=
I
$$i$$
i
Numerical answer
[src]
z1 = -0.707106781186548 + 0.707106781186548*i
z2 = 0.707106781186548 - 0.707106781186548*i
z2 = 0.707106781186548 - 0.707106781186548*i