F4 Nature of roots

ABCDis a parallelogram, where ÐABC = 30°, BC = 12cm　and DC = 8cm . If the length of each side of ABCD isdecreased by x cm, the area of the new parallelogram AEFG will be half old theoriginal area. Find the value of x correct to 1 decimal place.

Solution :
For parallelogram ABCD :
AB = DC = 8 cm

Area of parallelogram ABCD
= 2 ´ Area of ΔABC
= 2 ´ (1/2) ´ AB ´ BC ´ sinÐABC
= 8 ´ 12 ´ sin30° cm²
= 48 cm²


Similarly, area of parallelogram (in cm²):
2 ´ (1/2) ´ (8 - x) ´ (12 - x) ´ sin30° = (1/2) ´ 48
(8 - x)(12 - x) = 48
x² - 20x + 48 = 0
x = 10 - 2√13 or x = 10 + 2√13 (rejected)
x = 2.8 (to 1 decimal place)