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(n(n-1))/2=105
We move all terms to the left:
(n(n-1))/2-(105)=0
We multiply all the terms by the denominator
(n(n-1))-105*2=0
We calculate terms in parentheses: +(n(n-1)), so:We add all the numbers together, and all the variables
n(n-1)
We multiply parentheses
n^2-1n
Back to the equation:
+(n^2-1n)
(n^2-1n)-210=0
We get rid of parentheses
n^2-1n-210=0
a = 1; b = -1; c = -210;
Δ = b2-4ac
Δ = -12-4·1·(-210)
Δ = 841
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{841}=29$$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-1)-29}{2*1}=\frac{-28}{2} =-14 $$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1)+29}{2*1}=\frac{30}{2} =15 $
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