Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation 2*x^2-x+5=0
-
Equation e^x=1
-
Equation x^2+2=x+2
-
Equation 5=12-5(4x-1)
- Express {x} in terms of y where:
- -12*x-2*y=-15
- -4*x+4*y=-9
- -19*x-2*y=-6
- -2*x+2*y=4
- Canonical form:
- 2x^2-8x+11
-
Identical expressions
- two x^2-8x+ eleven
- 2x squared minus 8x plus 11
- two x squared minus 8x plus eleven
- 2x2-8x+11
- 2x²-8x+11
- 2x to the power of 2-8x+11
-
Similar expressions
- 2x^2-8x-11
- 2x^2+8x+11
2x^2-8x+11 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
$$a = 2$$
$$b = -8$$
$$c = 11$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = 2 + \frac{\sqrt{6} i}{2}$$
$$x_{2} = 2 - \frac{\sqrt{6} i}{2}$$
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
$$a = 2$$
$$b = -8$$
$$c = 11$$
, then
D = b^2 - 4 * a * c =
(-8)^2 - 4 * (2) * (11) = -24
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
$$x_{1} = 2 + \frac{\sqrt{6} i}{2}$$
$$x_{2} = 2 - \frac{\sqrt{6} i}{2}$$
Vieta's Theorem
rewrite the equation
$$\left(2 x^{2} - 8 x\right) + 11 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 4 x + \frac{11}{2} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -4$$
$$q = \frac{c}{a}$$
$$q = \frac{11}{2}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 4$$
$$x_{1} x_{2} = \frac{11}{2}$$
$$\left(2 x^{2} - 8 x\right) + 11 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 4 x + \frac{11}{2} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -4$$
$$q = \frac{c}{a}$$
$$q = \frac{11}{2}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 4$$
$$x_{1} x_{2} = \frac{11}{2}$$
Sum and product of roots
[src]
sum
___ ___
I*\/ 6 I*\/ 6
2 - ------- + 2 + -------
2 2
$$\left(2 - \frac{\sqrt{6} i}{2}\right) + \left(2 + \frac{\sqrt{6} i}{2}\right)$$
=
4
$$4$$
product
/ ___\ / ___\ | I*\/ 6 | | I*\/ 6 | |2 - -------|*|2 + -------| \ 2 / \ 2 /
$$\left(2 - \frac{\sqrt{6} i}{2}\right) \left(2 + \frac{\sqrt{6} i}{2}\right)$$
=
11/2
$$\frac{11}{2}$$
11/2
Rapid solution
[src]
___
I*\/ 6
x1 = 2 - -------
2
$$x_{1} = 2 - \frac{\sqrt{6} i}{2}$$
___
I*\/ 6
x2 = 2 + -------
2
$$x_{2} = 2 + \frac{\sqrt{6} i}{2}$$
x2 = 2 + sqrt(6)*i/2
Numerical answer
[src]
x1 = 2.0 - 1.22474487139159*i
x2 = 2.0 + 1.22474487139159*i
x2 = 2.0 + 1.22474487139159*i