(1) 2x²+5x+3 = 0
Using smart completing the square approach.
(x+(5/4))²+(3/2)-(5/4)²
= 0
(x+(5/4))² = (25/16)-(3/2)
(x+(5/4))² = (1/16)
x = -(5/4)±(1/4)
x = -1
Or
x = -1.5
Using factorization appro...
a² = 3²+4²
a = 5cm.
Where a is BC.
b² = 2.5²+4²-8*2.5cos(atan(3/4))
b = 2.5 cm.
c = 90-atan(3/4)
c = atan(4/3)
sin(atan(4/3)) = d/4
d = 3.2cm.
Where d is CN.
e = 5-3.2
e = 1.8cm.
Where e is BN....
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Let the side of the regular hexagon which is equal the side of the square be 1 unit.
a = ½(180-108)
a = 36°
tan36 = b/0.5
b = 0.363271264 unit.
Area Blue is;
0.5*0.363271264*0.5
= 0.090817816 s...
Let GD (inscribed square side) be a.
Calculating a.
2(a/(tan60))+a = 12
a = 12/((2/√(3))+1)
a = 5.56921938165 cm.
b = ½(12-5.56921938165)
b = 3.21539030918 cm.
Where b is BD = CE.
c = 3.21539030...
tana = (4/3)
a = atan(4/3)°
b = (135-atan(4/3))°
(c/sin45) = (6/(135-atan(4/3)))
c = 4.28571428571 units.
Therefore Area Blue is;
0.5*4.28571428571*6sin(atan(4/3))
= 10.28571428571 square units....
Notice;
The bigger square side is 1 unit.
Calculating the big square side.
Let it be a.
a² = 2(½)²
a = (1/√(2)) units.
Calculating the biggest square side.
Let it be b.
b² = (0.5)²+(1.5)²
b = ½√(...
Let the regular pentagon side length be 1 unit.
Single interior angle of the regular pentagon is;
⅕(180*3)
= 108°
a = 108-90
a = 18°
b = 180-18-108
b = 54°
(1/sin54) = c/sin108)
c = 1.1755705045...
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Let the square side be a.
Calculating a.
a²+(0.5a)² = 4²
a²+¼(a)² = 16
¼(5a²) = 16
5a² = 64
a = 8/√(5)
a = ⅕(8√(5)) units.
Half a is;
= ⅕(4√(5)) units.
tanb = ((4/√(5))/(8/√(5)))
b = atan(0.5)°...
