Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation -5x=30
-
Equation 3*x^2=18*x
-
Equation 16*x^2-9=0
-
Equation 0*x=5
- Express {x} in terms of y where:
- 10*x-18*y=-6
- 3*x-12*y=-14
- 16*x+11*y=-14
- -4*x+20*y=-14
- Factor polynomial:
- 2*x^2-x+1
- Factor squared:
- 2*x^2-x+1
-
Identical expressions
- two *x^ two -x+ one = zero
- 2 multiply by x squared minus x plus 1 equally 0
- two multiply by x to the power of two minus x plus one equally zero
- 2*x2-x+1=0
- 2*x²-x+1=0
- 2*x to the power of 2-x+1=0
- 2x^2-x+1=0
- 2x2-x+1=0
- 2*x^2-x+1=O
-
Similar expressions
- 2*x^2+x+1=0
- 2*x^2-x-1=0
2*x^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
$$a = 2$$
$$b = -1$$
$$c = 1$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = \frac{1}{4} + \frac{\sqrt{7} i}{4}$$
$$x_{2} = \frac{1}{4} - \frac{\sqrt{7} i}{4}$$
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
$$a = 2$$
$$b = -1$$
$$c = 1$$
, then
D = b^2 - 4 * a * c =
(-1)^2 - 4 * (2) * (1) = -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
$$x_{1} = \frac{1}{4} + \frac{\sqrt{7} i}{4}$$
$$x_{2} = \frac{1}{4} - \frac{\sqrt{7} i}{4}$$
Vieta's Theorem
rewrite the equation
$$\left(2 x^{2} - x\right) + 1 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{x}{2} + \frac{1}{2} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{1}{2}$$
$$q = \frac{c}{a}$$
$$q = \frac{1}{2}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{1}{2}$$
$$x_{1} x_{2} = \frac{1}{2}$$
$$\left(2 x^{2} - x\right) + 1 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{x}{2} + \frac{1}{2} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{1}{2}$$
$$q = \frac{c}{a}$$
$$q = \frac{1}{2}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{1}{2}$$
$$x_{1} x_{2} = \frac{1}{2}$$
Rapid solution
[src]
___
1 I*\/ 7
x1 = - - -------
4 4
$$x_{1} = \frac{1}{4} - \frac{\sqrt{7} i}{4}$$
___
1 I*\/ 7
x2 = - + -------
4 4
$$x_{2} = \frac{1}{4} + \frac{\sqrt{7} i}{4}$$
x2 = 1/4 + sqrt(7)*i/4
Sum and product of roots
[src]
sum
___ ___ 1 I*\/ 7 1 I*\/ 7 - - ------- + - + ------- 4 4 4 4
$$\left(\frac{1}{4} - \frac{\sqrt{7} i}{4}\right) + \left(\frac{1}{4} + \frac{\sqrt{7} i}{4}\right)$$
=
1/2
$$\frac{1}{2}$$
product
/ ___\ / ___\ |1 I*\/ 7 | |1 I*\/ 7 | |- - -------|*|- + -------| \4 4 / \4 4 /
$$\left(\frac{1}{4} - \frac{\sqrt{7} i}{4}\right) \left(\frac{1}{4} + \frac{\sqrt{7} i}{4}\right)$$
=
1/2
$$\frac{1}{2}$$
1/2
Numerical answer
[src]
x1 = 0.25 + 0.661437827766148*i
x2 = 0.25 - 0.661437827766148*i
x2 = 0.25 - 0.661437827766148*i
The graph