Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation x^4-2*x^2-15=0
-
Equation x^2-8x=0
-
Equation x^2-6x-27=0
-
Equation x^2=-3
- Express {x} in terms of y where:
- 8*x+1*y=4
- 20*x-16*y=7
- -3*x+19*y=16
- 8*x-16*y=14
-
Identical expressions
- two x- one /x+ six =2
- 2x minus 1 divide by x plus 6 equally 2
- two x minus one divide by x plus six equally 2
- 2x-1 divide by x+6=2
-
Similar expressions
- 2x-1/x-6=2
- 2x+1/x+6=2
2x-1/x+6=2 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$\left(2 x - \frac{1}{x}\right) + 6 = 2$$
Multiply the equation sides by the denominators:
and x
we get:
$$x \left(\left(2 x - \frac{1}{x}\right) + 6\right) = 2 x$$
$$2 x^{2} + 6 x - 1 = 2 x$$
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$2 x^{2} + 6 x - 1 = 2 x$$
to
$$2 x^{2} + 4 x - 1 = 0$$
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
$$a = 2$$
$$b = 4$$
$$c = -1$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = -1 + \frac{\sqrt{6}}{2}$$
$$x_{2} = - \frac{\sqrt{6}}{2} - 1$$
$$\left(2 x - \frac{1}{x}\right) + 6 = 2$$
Multiply the equation sides by the denominators:
and x
we get:
$$x \left(\left(2 x - \frac{1}{x}\right) + 6\right) = 2 x$$
$$2 x^{2} + 6 x - 1 = 2 x$$
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$2 x^{2} + 6 x - 1 = 2 x$$
to
$$2 x^{2} + 4 x - 1 = 0$$
This equation is of the form
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
$$a = 2$$
$$b = 4$$
$$c = -1$$
, then
D = b^2 - 4 * a * c =
(4)^2 - 4 * (2) * (-1) = 24
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = -1 + \frac{\sqrt{6}}{2}$$
$$x_{2} = - \frac{\sqrt{6}}{2} - 1$$
Rapid solution
[src]
___
\/ 6
x1 = -1 + -----
2
$$x_{1} = -1 + \frac{\sqrt{6}}{2}$$
___
\/ 6
x2 = -1 - -----
2
$$x_{2} = - \frac{\sqrt{6}}{2} - 1$$
x2 = -sqrt(6)/2 - 1
Sum and product of roots
[src]
sum
___ ___
\/ 6 \/ 6
-1 + ----- + -1 - -----
2 2
$$\left(- \frac{\sqrt{6}}{2} - 1\right) + \left(-1 + \frac{\sqrt{6}}{2}\right)$$
=
-2
$$-2$$
product
/ ___\ / ___\ | \/ 6 | | \/ 6 | |-1 + -----|*|-1 - -----| \ 2 / \ 2 /
$$\left(-1 + \frac{\sqrt{6}}{2}\right) \left(- \frac{\sqrt{6}}{2} - 1\right)$$
=
-1/2
$$- \frac{1}{2}$$
-1/2