The difference between the perimeters of two equilateral triangles is 9 cm, while the sum of their areas is $\frac{29}{4}\sqrt[ ]{3}$ cm². If x cm is the side of the smaller equilateral triangle, then represent the given information in the form of a quadratic equation.

Let the side of the smaller equilateral triangle = x cm 

∴ The perimeter of the smaller equilateral triangle = 3x cm 

The difference between the perimeters of the two equilateral triangles is given as 9 cm. 

∴Perimeter of the bigger equilateral triangle = (9 + 3x) cm 

⇒ Side of the bigger equilateral triangle = $\frac{9+3\mathrm{x}}{3}$ cm = (x + 3) cm 

Area of smaller equilateral triangle + Area of bigger equilateral triangle = $\frac{29}{4}\sqrt[ ]{3}$ cm² 

$\frac{\sqrt[ ]{3}}{4}{\mathrm{x}}^2+\frac{\sqrt[ ]{3}}{4}{(\mathrm{x}+3)}^2=\frac{29\sqrt[ ]{3}}{4}$ 

⇒ x² + (x + 3)² = 29 

⇒ x² + x² + 6x + 9 − 29 = 0 

⇒ 2x² + 6x − 20 = 0 

⇒ x² + 3x − 10 = 0 

Thus, the equation x² + 3x − 10 = 0 represents the given information.