Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation 15-x=8
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Equation x^2+3*x+9=0
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Equation x^2-16*x+64=0
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Equation x^2-6x-7=0
- Express {x} in terms of y where:
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Identical expressions
- z^ two +iz+ two = zero
- z squared plus iz plus 2 equally 0
- z to the power of two plus iz plus two equally zero
- z2+iz+2=0
- z²+iz+2=0
- z to the power of 2+iz+2=0
- z^2+iz+2=O
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Similar expressions
- z^2-iz+2=0
- z^2+iz-2=0
z^2+iz+2=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
$$a = 1$$
$$b = i$$
$$c = 2$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$z_{1} = i$$
Simplify
$$z_{2} = - 2 i$$
Simplify
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
$$a = 1$$
$$b = i$$
$$c = 2$$
, then
D = b^2 - 4 * a * c =
(i)^2 - 4 * (1) * (2) = -9
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
$$z_{1} = i$$
Simplify
$$z_{2} = - 2 i$$
Simplify
Vieta's Theorem
it is reduced quadratic equation
$$p z + z^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = i$$
$$q = \frac{c}{a}$$
$$q = 2$$
Vieta Formulas
$$z_{1} + z_{2} = - p$$
$$z_{1} z_{2} = q$$
$$z_{1} + z_{2} = - i$$
$$z_{1} z_{2} = 2$$
$$p z + z^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = i$$
$$q = \frac{c}{a}$$
$$q = 2$$
Vieta Formulas
$$z_{1} + z_{2} = - p$$
$$z_{1} z_{2} = q$$
$$z_{1} + z_{2} = - i$$
$$z_{1} z_{2} = 2$$
The graph
Sum and product of roots
[src]
sum
0 - 2*I + I
$$\left(0 - 2 i\right) + i$$
=
-I
$$- i$$
product
1*-2*I*I
$$i 1 \left(- 2 i\right)$$
=
2
$$2$$
2