Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation (x-4)^2+(x+9)^2=2x^2
-
Equation sqrt(x-1)=x
- Equation x^3+11x^2-9x-99=0
-
Equation 2x
- Express {x} in terms of y where:
- -10*x+18*y=6
- 9*x-18*y=6
- -1*x+19*y=-10
- 4*x+13*y=13
- Graphing y =:
- sqrt(x-1)
- Integral of d{x}:
-
sqrt(x-1)
- Derivative of:
-
sqrt(x-1)
-
Identical expressions
- sqrt(x- one)=x
- square root of (x minus 1) equally x
- square root of (x minus one) equally x
- √(x-1)=x
- sqrtx-1=x
-
Similar expressions
- 2^sqrt(x)+1-2^(sqrt(x)+1)-2^(sqrt(x)-1)=12
- x-sqrt(x)-12=0
- sqrt((x-1)/(x+1))*((x-1)/(x+1)-1)*(2/(x+1)+((x-1)/(x+1)-1)/(x-1)+2/(x-1))/(4*(x-1))=0
- sqrt(x+1)=x
sqrt(x-1)=x equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\sqrt{x - 1} = x$$
$$\sqrt{x - 1} = x$$
We raise the equation sides to 2-th degree
$$x - 1 = x^{2}$$
$$x - 1 = x^{2}$$
Transfer the right side of the equation left part with negative sign
$$- x^{2} + x - 1 = 0$$
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
$$a = -1$$
$$b = 1$$
$$c = -1$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = \frac{1}{2} - \frac{\sqrt{3} i}{2}$$
$$x_{2} = \frac{1}{2} + \frac{\sqrt{3} i}{2}$$
$$\sqrt{x - 1} = x$$
$$\sqrt{x - 1} = x$$
We raise the equation sides to 2-th degree
$$x - 1 = x^{2}$$
$$x - 1 = x^{2}$$
Transfer the right side of the equation left part with negative sign
$$- x^{2} + x - 1 = 0$$
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
$$a = -1$$
$$b = 1$$
$$c = -1$$
, then
D = b^2 - 4 * a * c =
(1)^2 - 4 * (-1) * (-1) = -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
$$x_{1} = \frac{1}{2} - \frac{\sqrt{3} i}{2}$$
$$x_{2} = \frac{1}{2} + \frac{\sqrt{3} i}{2}$$
Sum and product of roots
[src]
sum
___ ___ 1 I*\/ 3 1 I*\/ 3 - - ------- + - + ------- 2 2 2 2
$$\left(\frac{1}{2} - \frac{\sqrt{3} i}{2}\right) + \left(\frac{1}{2} + \frac{\sqrt{3} i}{2}\right)$$
=
1
$$1$$
product
/ ___\ / ___\ |1 I*\/ 3 | |1 I*\/ 3 | |- - -------|*|- + -------| \2 2 / \2 2 /
$$\left(\frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \left(\frac{1}{2} + \frac{\sqrt{3} i}{2}\right)$$
=
1
$$1$$
1
Rapid solution
[src]
___
1 I*\/ 3
x1 = - - -------
2 2
$$x_{1} = \frac{1}{2} - \frac{\sqrt{3} i}{2}$$
___
1 I*\/ 3
x2 = - + -------
2 2
$$x_{2} = \frac{1}{2} + \frac{\sqrt{3} i}{2}$$
x2 = 1/2 + sqrt(3)*i/2
Numerical answer
[src]
x1 = 0.5 - 0.866025403784439*i
x2 = 0.5 + 0.866025403784439*i
x2 = 0.5 + 0.866025403784439*i
The graph