Five number operations

For five whole numbers, if we add one to the first, multiply the second by the second, subtract three from the third, multiply the fourth by four, and divide the fifth by five, we get the same result each time. Find all five of the numbers that add up to 122.

Final Answer:

a =  15
b =  4
c =  19
d =  4
e =  80

Step-by-step explanation:

$$\begin{gathered} a+1=b^2 \\ a+1=c-3 \\ a+1=4d \\ a+1=e/5 \\ a+b+c+d+e=122 \\ (b^2-1)+b+(b^2+3)+(b^2/4)+(5\cdot b^2)=122 \\ 4(b^2-1)+4b+4(b^2+3)+b^2+5\cdot 4\cdot b^2=122\cdot 4 \\ 29b^2+4b=480 \\ 29b^2+4b-480=0 \\ p=29;q=4;r=-480 \\ D=q^2-4pr=4^2-4\cdot 29\cdot (-480)=55696 \\ D>0 \\ {b}_{1,2}=\frac{-q\pm \sqrt{D}}{2p}=\frac{-4\pm \sqrt{55696}}{58} \\ {b}_{1,2}=\frac{-4\pm 236}{58} \\ {b}_{1,2}=-0.068966\pm 4.068966 \\ {b}_1=4 \\ {b}_2=-4.137931034 \\ b=b_1=4 \\ {b}_2\notin N \\ a=b^2-1=4^2-1=15 \end{gathered}$$

Our quadratic equation calculator calculates it.

$b=4$

$c=b^2+3=4^2+3=19$

$d=b^2/4=4^2/4=4$

$e=5\cdot b^2=5\cdot 4^2=80$

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