Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation -4x+9=-19
-
Equation 3x^2-4x+2=0
- Equation (3b+5)x^2-2(b-1)x+2=0
- Equation 6x-3x=12
- Express {x} in terms of y where:
- 5*x-16*y=-12
- -18*x-14*y=-12
- -1*x+7*y=-9
- -9*x+4*y=-15
- Graphing y =:
-
3x^2-4x+2
-
Identical expressions
- 3x^ two -4x+ two = zero
- 3x squared minus 4x plus 2 equally 0
- 3x to the power of two minus 4x plus two equally zero
- 3x2-4x+2=0
- 3x²-4x+2=0
- 3x to the power of 2-4x+2=0
- 3x^2-4x+2=O
-
Similar expressions
- 3x^2+4x+2=0
- 3x^2-4x-2=0
3x^2-4x+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:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 3$$
$$b = -4$$
$$c = 2$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = \frac{2}{3} + \frac{\sqrt{2} i}{3}$$
$$x_{2} = \frac{2}{3} - \frac{\sqrt{2} i}{3}$$
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
$$a = 3$$
$$b = -4$$
$$c = 2$$
, then
D = b^2 - 4 * a * c =
(-4)^2 - 4 * (3) * (2) = -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
$$x_{1} = \frac{2}{3} + \frac{\sqrt{2} i}{3}$$
$$x_{2} = \frac{2}{3} - \frac{\sqrt{2} i}{3}$$
Vieta's Theorem
rewrite the equation
$$\left(3 x^{2} - 4 x\right) + 2 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{4 x}{3} + \frac{2}{3} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{4}{3}$$
$$q = \frac{c}{a}$$
$$q = \frac{2}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{4}{3}$$
$$x_{1} x_{2} = \frac{2}{3}$$
$$\left(3 x^{2} - 4 x\right) + 2 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{4 x}{3} + \frac{2}{3} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{4}{3}$$
$$q = \frac{c}{a}$$
$$q = \frac{2}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{4}{3}$$
$$x_{1} x_{2} = \frac{2}{3}$$
Rapid solution
[src]
___
2 I*\/ 2
x1 = - - -------
3 3
$$x_{1} = \frac{2}{3} - \frac{\sqrt{2} i}{3}$$
___
2 I*\/ 2
x2 = - + -------
3 3
$$x_{2} = \frac{2}{3} + \frac{\sqrt{2} i}{3}$$
x2 = 2/3 + sqrt(2)*i/3
Sum and product of roots
[src]
sum
___ ___ 2 I*\/ 2 2 I*\/ 2 - - ------- + - + ------- 3 3 3 3
$$\left(\frac{2}{3} - \frac{\sqrt{2} i}{3}\right) + \left(\frac{2}{3} + \frac{\sqrt{2} i}{3}\right)$$
=
4/3
$$\frac{4}{3}$$
product
/ ___\ / ___\ |2 I*\/ 2 | |2 I*\/ 2 | |- - -------|*|- + -------| \3 3 / \3 3 /
$$\left(\frac{2}{3} - \frac{\sqrt{2} i}{3}\right) \left(\frac{2}{3} + \frac{\sqrt{2} i}{3}\right)$$
=
2/3
$$\frac{2}{3}$$
2/3
Numerical answer
[src]
x1 = 0.666666666666667 - 0.471404520791032*i
x2 = 0.666666666666667 + 0.471404520791032*i
x2 = 0.666666666666667 + 0.471404520791032*i
The graph