Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation x^2-6x+9=0
-
Equation 2*x+2=-3
-
Equation 2x+2=-3
-
Equation (x-11)^4=(x+3)^4
- Express {x} in terms of y where:
- -10*x-18*y=15
- 12*x+8*y=18
- 9*x+14*y=-10
- 18*x+1*y=-11
-
Identical expressions
- z^ two -3z-1 zero i=0
- z squared minus 3z minus 10i equally 0
- z to the power of two minus 3z minus 1 zero i equally 0
- z2-3z-10i=0
- z²-3z-10i=0
- z to the power of 2-3z-10i=0
- z^2-3z-10i=O
-
Similar expressions
- z^2-3z+10i=0
- z^2+3z-10i=0
z^2-3z-10i=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
$$a = 1$$
$$b = -3$$
$$c = - 10 i$$
, then
The equation has two roots.
or
$$z_{1} = 4 + 2 i$$
$$z_{2} = -1 - 2 i$$
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
$$a = 1$$
$$b = -3$$
$$c = - 10 i$$
, then
D = b^2 - 4 * a * c =
(-3)^2 - 4 * (1) * (-10*i) = 9 + 40*i
The equation has two roots.
z1 = (-b + sqrt(D)) / (2*a)
z2 = (-b - sqrt(D)) / (2*a)
or
$$z_{1} = 4 + 2 i$$
$$z_{2} = -1 - 2 i$$
Vieta's Theorem
it is reduced quadratic equation
$$p z + q + z^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -3$$
$$q = \frac{c}{a}$$
$$q = - 10 i$$
Vieta Formulas
$$z_{1} + z_{2} = - p$$
$$z_{1} z_{2} = q$$
$$z_{1} + z_{2} = 3$$
$$z_{1} z_{2} = - 10 i$$
$$p z + q + z^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -3$$
$$q = \frac{c}{a}$$
$$q = - 10 i$$
Vieta Formulas
$$z_{1} + z_{2} = - p$$
$$z_{1} z_{2} = q$$
$$z_{1} + z_{2} = 3$$
$$z_{1} z_{2} = - 10 i$$
The graph
Sum and product of roots
[src]
sum
-1 - 2*I + 4 + 2*I
$$\left(-1 - 2 i\right) + \left(4 + 2 i\right)$$
=
3
$$3$$
product
(-1 - 2*I)*(4 + 2*I)
$$\left(-1 - 2 i\right) \left(4 + 2 i\right)$$
=
-10*I
$$- 10 i$$
-10*i
Rapid solution
[src]
z1 = -1 - 2*I
$$z_{1} = -1 - 2 i$$
z2 = 4 + 2*I
$$z_{2} = 4 + 2 i$$
z2 = 4 + 2*i