Which of the following conclusions is true about the statement below x^2=sqrtx

None.

To determine which of the following conclusions is true about the statement x^2 = sqrt(x), we need to analyze the equation and its properties.

First, let's consider the meaning of each side of the equation:

- x^2: This represents x raised to the power of 2, which is the same as x multiplied by itself.
- sqrt(x): This represents the square root of x, which is a value that, when multiplied by itself, equals x.

Now, let's solve the equation x^2 = sqrt(x) to determine the possible conclusions:

1. There are no real solutions for x: To confirm this conclusion, we need to find the values of x that satisfy the equation. By squaring both sides of the equation, we get x^4 = x. Rearranging, we have x^4 - x = 0. However, this equation cannot be solved as it is not easily factorizable. Therefore, we cannot conclude that there are no real solutions for x.

2. x = 0 is the only solution: To verify this conclusion, we substitute x = 0 into the original equation x^2 = sqrt(x). We have 0^2 = sqrt(0), which simplifies to 0 = 0. Since the equation is satisfied, x = 0 is indeed a valid solution. However, we should check if there are additional solutions.

3. There are other real solutions besides x = 0: To explore this conclusion, we can examine the graphical representation of the equation. By plotting y = x^2 and y = sqrt(x) on a graphing tool, we can observe the points where the two curves intersect. If there are any additional intersections other than the point (0,0), this would indicate the presence of other real solutions.

Upon analyzing the graph, we can observe that there is indeed one additional intersection point, near (1,1). Therefore, we conclude that apart from x = 0, there is one more real solution to the equation x^2 = sqrt(x).

In summary, the correct conclusion is: "There is one additional real solution besides x = 0." This conclusion is based on both solving the equation algebraically and analyzing the graphical representation.