Mister Exam
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How to use it?
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Identical expressions
- 3x^ two -x+ two = zero
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Similar expressions
- 3x^2-x-2=0
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3x^2-x+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
$$a = 3$$
$$b = -1$$
$$c = 2$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = \frac{1}{6} + \frac{\sqrt{23} i}{6}$$
$$x_{2} = \frac{1}{6} - \frac{\sqrt{23} i}{6}$$
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
$$a = 3$$
$$b = -1$$
$$c = 2$$
, then
D = b^2 - 4 * a * c =
(-1)^2 - 4 * (3) * (2) = -23
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
$$x_{1} = \frac{1}{6} + \frac{\sqrt{23} i}{6}$$
$$x_{2} = \frac{1}{6} - \frac{\sqrt{23} i}{6}$$
Vieta's Theorem
rewrite the equation
$$\left(3 x^{2} - x\right) + 2 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{x}{3} + \frac{2}{3} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{1}{3}$$
$$q = \frac{c}{a}$$
$$q = \frac{2}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{1}{3}$$
$$x_{1} x_{2} = \frac{2}{3}$$
$$\left(3 x^{2} - x\right) + 2 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{x}{3} + \frac{2}{3} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{1}{3}$$
$$q = \frac{c}{a}$$
$$q = \frac{2}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{1}{3}$$
$$x_{1} x_{2} = \frac{2}{3}$$
Rapid solution
[src]
____
1 I*\/ 23
x1 = - - --------
6 6
$$x_{1} = \frac{1}{6} - \frac{\sqrt{23} i}{6}$$
____
1 I*\/ 23
x2 = - + --------
6 6
$$x_{2} = \frac{1}{6} + \frac{\sqrt{23} i}{6}$$
x2 = 1/6 + sqrt(23)*i/6
Sum and product of roots
[src]
sum
____ ____ 1 I*\/ 23 1 I*\/ 23 - - -------- + - + -------- 6 6 6 6
$$\left(\frac{1}{6} - \frac{\sqrt{23} i}{6}\right) + \left(\frac{1}{6} + \frac{\sqrt{23} i}{6}\right)$$
=
1/3
$$\frac{1}{3}$$
product
/ ____\ / ____\ |1 I*\/ 23 | |1 I*\/ 23 | |- - --------|*|- + --------| \6 6 / \6 6 /
$$\left(\frac{1}{6} - \frac{\sqrt{23} i}{6}\right) \left(\frac{1}{6} + \frac{\sqrt{23} i}{6}\right)$$
=
2/3
$$\frac{2}{3}$$
2/3
Numerical answer
[src]
x1 = 0.166666666666667 + 0.799305253885453*i
x2 = 0.166666666666667 - 0.799305253885453*i
x2 = 0.166666666666667 - 0.799305253885453*i
The graph