Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation x/4+x=4
-
Equation x^2+11=0
- Equation x+y=5
-
Equation 1*(x-1)*(x-3)=0
- Express {x} in terms of y where:
- -11*x-3*y=14
- -1*x-17*y=-12
- 20*x+18*y=-9
- -2*x+20*y=-2
- Factor polynomial:
- 3*x^2+2*x+1
- Graphing y =:
- 3*x^2+2*x+1
- Factor squared:
- 3*x^2+2*x+1
-
Identical expressions
- three *x^ two + two *x+ one = zero
- 3 multiply by x squared plus 2 multiply by x plus 1 equally 0
- three multiply by x to the power of two plus two multiply by x plus one equally zero
- 3*x2+2*x+1=0
- 3*x²+2*x+1=0
- 3*x to the power of 2+2*x+1=0
- 3x^2+2x+1=0
- 3x2+2x+1=0
- 3*x^2+2*x+1=O
-
Similar expressions
- 3*x^2-2*x+1=0
- 3*x^2+2*x-1=0
3*x^2+2*x+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:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 3$$
$$b = 2$$
$$c = 1$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = - \frac{1}{3} + \frac{\sqrt{2} i}{3}$$
$$x_{2} = - \frac{1}{3} - \frac{\sqrt{2} i}{3}$$
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
$$a = 3$$
$$b = 2$$
$$c = 1$$
, then
D = b^2 - 4 * a * c =
(2)^2 - 4 * (3) * (1) = -8
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
$$x_{1} = - \frac{1}{3} + \frac{\sqrt{2} i}{3}$$
$$x_{2} = - \frac{1}{3} - \frac{\sqrt{2} i}{3}$$
Vieta's Theorem
rewrite the equation
$$\left(3 x^{2} + 2 x\right) + 1 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{2 x}{3} + \frac{1}{3} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{2}{3}$$
$$q = \frac{c}{a}$$
$$q = \frac{1}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{2}{3}$$
$$x_{1} x_{2} = \frac{1}{3}$$
$$\left(3 x^{2} + 2 x\right) + 1 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{2 x}{3} + \frac{1}{3} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{2}{3}$$
$$q = \frac{c}{a}$$
$$q = \frac{1}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{2}{3}$$
$$x_{1} x_{2} = \frac{1}{3}$$
Rapid solution
[src]
___
1 I*\/ 2
x1 = - - - -------
3 3
$$x_{1} = - \frac{1}{3} - \frac{\sqrt{2} i}{3}$$
___
1 I*\/ 2
x2 = - - + -------
3 3
$$x_{2} = - \frac{1}{3} + \frac{\sqrt{2} i}{3}$$
x2 = -1/3 + sqrt(2)*i/3
Sum and product of roots
[src]
sum
___ ___ 1 I*\/ 2 1 I*\/ 2 - - - ------- + - - + ------- 3 3 3 3
$$\left(- \frac{1}{3} - \frac{\sqrt{2} i}{3}\right) + \left(- \frac{1}{3} + \frac{\sqrt{2} i}{3}\right)$$
=
-2/3
$$- \frac{2}{3}$$
product
/ ___\ / ___\ | 1 I*\/ 2 | | 1 I*\/ 2 | |- - - -------|*|- - + -------| \ 3 3 / \ 3 3 /
$$\left(- \frac{1}{3} - \frac{\sqrt{2} i}{3}\right) \left(- \frac{1}{3} + \frac{\sqrt{2} i}{3}\right)$$
=
1/3
$$\frac{1}{3}$$
1/3
Numerical answer
[src]
x1 = -0.333333333333333 + 0.471404520791032*i
x2 = -0.333333333333333 - 0.471404520791032*i
x2 = -0.333333333333333 - 0.471404520791032*i
The graph