Saturday, February 11, 2023

Sum and Product Problem with the Casio fx-4000P

Sum and Product Problem with the Casio fx-4000P



The Sum and Product Problem


For the numbers a and b with the sum s, and product p:


a + b = s

a * b = p


Let's find a solution for a and b:


a + b = s

b = s - a


a * b = p

a * (s - a) = p

a * s - a^2 = p

0 = a^2 - a * s + p


a = (s ± √(s^2 - 4 * p))/2


Then:


b = s - a

= s -  (s ± √(s^2 - 4 * p))/2

= (2*s)/2 -  (s ± √(s^2 - 4 * p))/2

= (2*s -s ±√(s^2 - 4 * p))/2

=  (s ± √(s^2 - 4 * p))/2


Note that addition and multiplication are communitive.  


Without loss of generality, let:


a = (s + √(s^2 - 4 * p))/2


b = (s - √(s^2 - 4 * p))/2


Verification:


a + b 

= (s + √(s^2 - 4*p))/2 + (s - √(s^2 - 4*p))/2

= (s + √(s^2 - 4*p) + s - √(s^2 - 4*p))/2

= (2 * s)/2

= s


a * b

= (s + √(s^2 - 4 * p))/2 * (s + √(s^2 - 4 * p))/2

= (s/2)^2 - (√(s^2 - 4 * p)/2)^2

= s^2/4 - (s^2 - 4 *p)/4 

= (s^2 - s^2 + 4 * p)/4

= (4 * p)/4

= p



Casio fx-4000P Program:  Sum and Product Problem


This program can be adopted to many programming and graphing calculators. 


Program:


"A+B=":?→S:

"AB=":?→P:

(S+√(S²-4P))÷2→A◢

S-A→B



Examples


S:  Sum

P:  Product


Example 1:

S = 12, P =32;

Results: 8, 4


Example 2:

S = 8, P = 15;

Results: 5, 3



Enjoy!  Until next time,


Eddie


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