Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation e^x=0
-
Equation x^4-1=0
- Equation x+y+2=0
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Equation x^2-3*x+3=0
- Express {x} in terms of y where:
- 2*x+2*y=-6
- 10*x-2*y=0
- -9*x-4*y=-20
- 4*x-7*y=4
- Factor squared:
- x^2-3*x+3
-
Identical expressions
- x^ two - three *x+ three = zero
- x squared minus 3 multiply by x plus 3 equally 0
- x to the power of two minus three multiply by x plus three equally zero
- x2-3*x+3=0
- x²-3*x+3=0
- x to the power of 2-3*x+3=0
- x^2-3x+3=0
- x2-3x+3=0
- x^2-3*x+3=O
-
Similar expressions
- x^2-3*x-3=0
- 1/(x^2-3x+3)-2/(x^2-3x+4)=6/(x^2-3x+5)
- x^2+3*x+3=0
x^2-3*x+3=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
$$a = 1$$
$$b = -3$$
$$c = 3$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = \frac{3}{2} + \frac{\sqrt{3} i}{2}$$
$$x_{2} = \frac{3}{2} - \frac{\sqrt{3} i}{2}$$
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
$$a = 1$$
$$b = -3$$
$$c = 3$$
, then
D = b^2 - 4 * a * c =
(-3)^2 - 4 * (1) * (3) = -3
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
$$x_{1} = \frac{3}{2} + \frac{\sqrt{3} i}{2}$$
$$x_{2} = \frac{3}{2} - \frac{\sqrt{3} i}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -3$$
$$q = \frac{c}{a}$$
$$q = 3$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 3$$
$$x_{1} x_{2} = 3$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -3$$
$$q = \frac{c}{a}$$
$$q = 3$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 3$$
$$x_{1} x_{2} = 3$$
Rapid solution
[src]
___
3 I*\/ 3
x1 = - - -------
2 2
$$x_{1} = \frac{3}{2} - \frac{\sqrt{3} i}{2}$$
___
3 I*\/ 3
x2 = - + -------
2 2
$$x_{2} = \frac{3}{2} + \frac{\sqrt{3} i}{2}$$
x2 = 3/2 + sqrt(3)*i/2
Sum and product of roots
[src]
sum
___ ___ 3 I*\/ 3 3 I*\/ 3 - - ------- + - + ------- 2 2 2 2
$$\left(\frac{3}{2} - \frac{\sqrt{3} i}{2}\right) + \left(\frac{3}{2} + \frac{\sqrt{3} i}{2}\right)$$
=
3
$$3$$
product
/ ___\ / ___\ |3 I*\/ 3 | |3 I*\/ 3 | |- - -------|*|- + -------| \2 2 / \2 2 /
$$\left(\frac{3}{2} - \frac{\sqrt{3} i}{2}\right) \left(\frac{3}{2} + \frac{\sqrt{3} i}{2}\right)$$
=
3
$$3$$
3
Numerical answer
[src]
x1 = 1.5 - 0.866025403784439*i
x2 = 1.5 + 0.866025403784439*i
x2 = 1.5 + 0.866025403784439*i
The graph