To find the value of \( x \) for the triangle with sides \( x \), \( 2x \), and \( x^2 \) and a perimeter of 10 cm, we can start by expressing the perimeter in terms of \( x \).
The perimeter \( P \) of the triangle is given by:
\[ P = x + 2x + x^2 = 10 \]
Simplifying this equation, we have:
\[ 3x + x^2 = 10 \]
Rearranging the equation, we get:
\[ x^2 + 3x - 10 = 0 \]
Next, we will solve this quadratic equation using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1 \), \( b = 3 \), and \( c = -10 \).
Calculating the discriminant:
\[ b^2 - 4ac = 3^2 - 4(1)(-10) = 9 + 40 = 49 \]
Now substituting into the quadratic formula:
\[ x = \frac{-3 \pm \sqrt{49}}{2 \cdot 1} = \frac{-3 \pm 7}{2} \]
Calculating the two possible values for \( x \):
- \( x = \frac{-3 + 7}{2} = \frac{4}{2} = 2 \)
- \( x = \frac{-3 - 7}{2} = \frac{-10}{2} = -5 \) (not valid since \( x \) must be positive)
Thus, the only valid solution is:
\[ x = 2 \]
We can verify that the sides of the triangle are:
- \( x = 2 \)
- \( 2x = 4 \)
- \( x^2 = 4 \)
Now, we check the perimeter:
\[ 2 + 4 + 4 = 10 \text{ cm} \]
Since these values satisfy the perimeter and confirm that they fulfill the triangle inequality (each side must be less than the sum of the other two sides), we conclude that:
\[ \text{The value of } x \text{ is } \boxed{2}. \]