The difference between the perimeters of two equilateral triangles is 9 cm, while the sum of their areas is $\frac{29}{4}\sqrt{3} {\mathrm{cm}}^2$. If x cm is the side of the smaller equilateral triangle, then which quadratic equation represents the given information? 

Side of the smaller equilateral triangle = x cm

∴ 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 = $\frac{9+3x}{3} cm$= (x + 3) cm

Area of smaller equilateral triangle + Area of bigger equilateral triangle = $\frac{29}{4}\sqrt{3} {\mathrm{cm}}^2$ 

$\therefore \frac{\sqrt{3}}{4}{x}^2+\frac{\sqrt{3}}{4}{\left(x+3\right)}^2=\frac{29\sqrt{3}}{4}$

⇒ 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.