x + 1 = x^0.5?

Question

Golden ratio's little brother.

Answer 1

(x + 1)² = x
x² + 2x + 1 = x
x² + x + 1 = 0
x = [-1 ± √(1 - 4) ] ∕ 2
x = [ -1 ± √(- 3) ] ∕ 2
x = [-1 ± i √(3) ] ∕ 2

Answer 2

x+1=x^0.5
(x+1)^2=x
x^2+2x+1=x
x(x+1)+1=0

Draw the diagram and then you find out how this polynom follows the vales of x.
It is 1 at x =0.
Also the x= -1 gives the value 1 of this polynom.
So what is going on in the areas of x-value between 0 and -1.
Be patient and calculate the diagram, and you will understand the problem


.

Answer 3

Square both sides (^2)
x^2 + 1^2 = x^ 1/2*2
x^2 + 1 = x
x^2 -x +1 = 0
no this equation can not be solved by real numbers
Here x must have two answers with imagine part
there for
x1 = 3/2 + i
x2 = -1/2 + i

Answer 4

x+1=x^0.5 condition x>=0
(x+1)^2=x
x^2+2x+1=x
x^2+x+1=0
because x>=0 so x^2>=0
->x^2+x+1>=1>0
->^^

Answer 5

It should be "-1".
x - √x - 1 = 0, x > 0

Solve for √x with quadratic formula,
√x = (1+√5)/2, since x > 0

Square both sides,
x = (1+5+2√5)/4 = (3+√5)/2

Answer 6

To add a different approach:
use a graphical calculator to plot:
y = x+1 and y = x^0.5
(for x >=0)

These graphs do not intersect, and so there is no real solution.

Answer 7

x + 1 = x^0.5
x - x^0.5 + 1 = 0

Let y=x^0.5

y^2 - y + 1 = 0
Discriminant = (-1)^2 - 4(1)(1) = -3

Hence no solution.

Answer 8

x>=0

(x+1)^2 = x
x^2 + x +1 =0 --> no solution

Answer 9

x + 1 = x^0.5
y =x+1 -x^0.5

Domain = [0, +infinity[

minimum value is 1
function does not intersect the x axis

This equation has no real solutions

Answer 10

x + 1 = x power (0.5) (used ln both sides) (*)

ln(x + 1) = 0.5 ln(x) (used e power x both sides)

x + 1 = 0.5

x = - 0.5 (no solution, because x of (*) must
be always positive for real
numbers.)