Find the value of whole root of 6+root 6+root 6+....... .
x=√(6+√(6+√(6+...
x^2 = 6+√(6+√(6+√(6+...
x^2 = 6+x
x^2-x-6 = 0
(x-3)(x+2)=0
x = 3
To find the value of the whole root of the given expression, we can start by assigning a variable to the expression. Let's say:
x = √(6 + √(6 + √(6 + ...)))
Next, we can substitute the expression itself into the equation, like this:
x = √(6 + x)
Now, we can solve this equation by squaring both sides:
x^2 = 6 + x
Rearranging the equation to the quadratic form:
x^2 - x - 6 = 0
The equation can be factored as:
(x - 3)(x + 2) = 0
From this, we find two possible solutions:
x - 3 = 0 -> x = 3
x + 2 = 0 -> x = -2
Since the whole root represents a positive value, the solution x = -2 is extraneous and can be disregarded.
Therefore, the value of the whole root of 6 + √(6 + √(6 + ...)) is 3.
To find the value of the whole root of 6+root 6+root 6+..., we can use algebraic manipulation and solve for the value.
Let's denote the given expression as 'x'.
x = 6 + √(6 + √(6 + √(6 + ...)))
Notice that the expression inside the square root repeats indefinitely. We can represent this repetition as:
x = 6 + √(x)
Now, we can solve this equation using algebraic techniques.
Subtracting x from both sides:
x - 6 = √(x)
Squaring both sides to eliminate the square root:
(x - 6)^2 = x
Expanding the left side:
x^2 - 12x + 36 = x
Moving all terms to one side:
x^2 - 13x + 36 = 0
Now, we can solve this quadratic equation. Factoring or using the quadratic formula will give us the two solutions. In this case, the equation can be factored as:
(x - 4)(x - 9) = 0
Setting each factor equal to zero:
x - 4 = 0 or x - 9 = 0
Solving for x:
x = 4 or x = 9
Since the value of x cannot be negative, we discard the solution x = 4. Therefore, the value of the whole root of the given expression is:
x = 9.