Diagonals

What x-gon has 54 diagonals?

Result

x =  12

Solution:

 −0.5n2+1.5n+54=0 0.5n2−1.5n−54=0  a=0.5;b=−1.5;c=−54 D=b2−4ac=1.52−4⋅0.5⋅(−54)=110.25 D>0  n1,2=−b±D2a=1.5±110.251 n1,2=1.5±10.5 n1=12 n2=−9   Factored form of the equation:  0.5(n−12)(n+9)=0  x>0  x=n1=12 \ \\ -0.5n^2 +1.5n +54=0 \ \\ 0.5n^2 -1.5n -54=0 \ \\ \ \\ a=0.5; b=-1.5; c=-54 \ \\ D=b^2 - 4ac=1.5^2 - 4\cdot 0.5 \cdot (-54)=110.25 \ \\ D>0 \ \\ \ \\ n_{1,2}=\dfrac{ -b \pm \sqrt{ D } }{ 2a }=\dfrac{ 1.5 \pm \sqrt{ 110.25 } }{ 1 } \ \\ n_{1,2}=1.5 \pm 10.5 \ \\ n_{1}=12 \ \\ n_{2}=-9 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ 0.5 (n -12) (n +9)=0 \ \\ \ \\ x>0 \ \\ \ \\ x=n_{1}=12

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