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n(n-3)/2=90
We move all terms to the left:
n(n-3)/2-(90)=0
We multiply all the terms by the denominator
n(n-3)-90*2=0
We add all the numbers together, and all the variables
n(n-3)-180=0
We multiply parentheses
n^2-3n-180=0
a = 1; b = -3; c = -180;
Δ = b2-4ac
Δ = -32-4·1·(-180)
Δ = 729
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$$\sqrt{\Delta}=\sqrt{729}=27$$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-3)-27}{2*1}=\frac{-24}{2} =-12 $$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-3)+27}{2*1}=\frac{30}{2} =15 $
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