Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation 9-4x=3x-40
-
Equation x^2=8
-
Equation x+4=0
-
Equation x^2+6x+9=0
- Express {x} in terms of y where:
- 18*x+12*y=18
- -11*x-3*y=14
- -20*x-13*y=15
- 12*x-3*y=-2
- Factor squared:
- x^2-x+2
- Factor polynomial:
- x^2-x+2
- Derivative of:
-
x^2-x+2
-
Identical expressions
- x^ two -x+ two
- x squared minus x plus 2
- x to the power of two minus x plus two
- x2-x+2
- x²-x+2
- x to the power of 2-x+2
-
Similar expressions
- x^2+x+2
- x^2-x-2
x^2-x+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
$$a = 1$$
$$b = -1$$
$$c = 2$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = \frac{1}{2} + \frac{\sqrt{7} i}{2}$$
$$x_{2} = \frac{1}{2} - \frac{\sqrt{7} i}{2}$$
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
$$a = 1$$
$$b = -1$$
$$c = 2$$
, then
D = b^2 - 4 * a * c =
(-1)^2 - 4 * (1) * (2) = -7
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
$$x_{1} = \frac{1}{2} + \frac{\sqrt{7} i}{2}$$
$$x_{2} = \frac{1}{2} - \frac{\sqrt{7} i}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = 2$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 1$$
$$x_{1} x_{2} = 2$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = 2$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 1$$
$$x_{1} x_{2} = 2$$
Sum and product of roots
[src]
sum
___ ___ 1 I*\/ 7 1 I*\/ 7 - - ------- + - + ------- 2 2 2 2
$$\left(\frac{1}{2} - \frac{\sqrt{7} i}{2}\right) + \left(\frac{1}{2} + \frac{\sqrt{7} i}{2}\right)$$
=
1
$$1$$
product
/ ___\ / ___\ |1 I*\/ 7 | |1 I*\/ 7 | |- - -------|*|- + -------| \2 2 / \2 2 /
$$\left(\frac{1}{2} - \frac{\sqrt{7} i}{2}\right) \left(\frac{1}{2} + \frac{\sqrt{7} i}{2}\right)$$
=
2
$$2$$
2
Rapid solution
[src]
___
1 I*\/ 7
x1 = - - -------
2 2
$$x_{1} = \frac{1}{2} - \frac{\sqrt{7} i}{2}$$
___
1 I*\/ 7
x2 = - + -------
2 2
$$x_{2} = \frac{1}{2} + \frac{\sqrt{7} i}{2}$$
x2 = 1/2 + sqrt(7)*i/2
Numerical answer
[src]
x1 = 0.5 + 1.3228756555323*i
x2 = 0.5 - 1.3228756555323*i
x2 = 0.5 - 1.3228756555323*i