Given the lengths of the sides of the triangle \( x \), \( 2x \), and \( x^2 \), we can write the equation for the perimeter:
\[ x + 2x + x^2 = 10 \]
Simplifying this, we have:
\[ 3x + x^2 = 10 \]
Rearranging the equation gives us:
\[ x^2 + 3x - 10 = 0 \]
Now we can solve this quadratic equation using the quadratic formula, where \( a = 1 \), \( b = 3 \), and \( c = -10 \):
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot (-10)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ x = \frac{-3 \pm \sqrt{9 + 40}}{2} \]
\[ x = \frac{-3 \pm \sqrt{49}}{2} \]
\[ x = \frac{-3 \pm 7}{2} \]
This gives us two potential solutions:
- \( x = \frac{4}{2} = 2 \)
- \( x = \frac{-10}{2} = -5 \)
Since \( x \) must be a positive length, we take:
\[ x = 2 \]
Now, let's confirm that these values satisfy the triangle inequality and the perimeter:
- The sides of the triangle are: \( x = 2 \), \( 2x = 4 \), and \( x^2 = 4 \).
- The perimeter is:
\[ 2 + 4 + 4 = 10 \text{ cm (correct)} \]
-
Checking triangle inequality:
- \( 2 + 4 > 4 \) (true)
- \( 2 + 4 > 4 \) (true)
- \( 4 + 4 > 2 \) (true)
All conditions are satisfied, thus the value of \( x \) is:
\[ \boxed{2} \text{ cm} \]