Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation (x+30)*5=365
-
Equation x/3/7=9/14
-
Equation 4x+8=2x
-
Equation log(x)-3+1/x=0
- Express {x} in terms of y where:
- 8*x-7*y=-1
- -4*x-1*y=-6
- 9*x-14*y=-6
- -12*x+19*y=2
- Graphing y =:
- 3x^2+2
- Derivative of:
-
3x^2+2
- Integral of d{x}:
- 3x^2+2
-
Identical expressions
- 3x^ two + two
- 3x squared plus 2
- 3x to the power of two plus two
- 3x2+2
- 3x²+2
- 3x to the power of 2+2
-
Similar expressions
- 3x^2-2
3x^2+2 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:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 3$$
$$b = 0$$
$$c = 2$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = \frac{\sqrt{6} i}{3}$$
$$x_{2} = - \frac{\sqrt{6} i}{3}$$
a*x^2 + b*x + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 3$$
$$b = 0$$
$$c = 2$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (3) * (2) = -24
Because D<0, then the equation
has no real roots,
but complex roots is exists.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{\sqrt{6} i}{3}$$
$$x_{2} = - \frac{\sqrt{6} i}{3}$$
Vieta's Theorem
rewrite the equation
$$3 x^{2} + 2 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{2}{3} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = \frac{2}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 0$$
$$x_{1} x_{2} = \frac{2}{3}$$
$$3 x^{2} + 2 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{2}{3} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = \frac{2}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 0$$
$$x_{1} x_{2} = \frac{2}{3}$$
Rapid solution
[src]
___
-I*\/ 6
x1 = ---------
3
$$x_{1} = - \frac{\sqrt{6} i}{3}$$
___
I*\/ 6
x2 = -------
3
$$x_{2} = \frac{\sqrt{6} i}{3}$$
x2 = sqrt(6)*i/3
Sum and product of roots
[src]
sum
___ ___
I*\/ 6 I*\/ 6
- ------- + -------
3 3
$$- \frac{\sqrt{6} i}{3} + \frac{\sqrt{6} i}{3}$$
=
0
$$0$$
product
___ ___
-I*\/ 6 I*\/ 6
---------*-------
3 3
$$- \frac{\sqrt{6} i}{3} \frac{\sqrt{6} i}{3}$$
=
2/3
$$\frac{2}{3}$$
2/3