Math
posted by LP .
A Quadratic Word Problem.
The difference between two positive numbers is 3. The sum of the squares of the numbers is 89. Find the numbers.

a = first number
b = second number
a  b = 3 Add b to both sides
a  b + b = 3 + b
a = b + 3
a ^ 2 + b ^ 2 = 89
( b + 3 ) ^ 2 + b ^ 2 = 89
b ^ 2 + 2 * b * 3 + 3 ^ 2 + b ^ 2 = 89
b ^ 2 + 6 b + 9 + b ^ 2 = 89
2 b ^ 2 + 6 b + 9 = 89 Divide both sides by 2
b ^ 2 + 3 b + 9 / 2 = 89 / 2 Subtract 9 / 2 to both sides
b ^ 2 + 3 b + 9 / 2  9 / 2 = 89 / 2  9 / 2
b ^ 2 + 3 b = 80 / 2
b ^ 2 + 3 b = 40 Add 9 / 4 to both sides
b ^ 2 + 3 b + 9 / 4 = 40 + 9 / 4
b ^ 2 + 3 b + 9 / 4 = 160 / 4 + 9 / 4
b ^ 2 + 3 b + 9 / 4 = 169 / 4
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Remark :
b ^ 2 + 3 b + 9 / 4 = ( b + 3 / 2 ) ^ 2
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( b + 3 / 2 ) ^ 2 = 169 / 4 take sqrt to both sides
b + 3 / 2 = 13 / 2 Subtract 3 / 2 to both sides
b + 3 / 2  3 / 2 = 13 / 2  3 / 2
b = 10 / 2 = 5
a = b + 3 = 5 + 3 = 8
Nuumbers are :
5 and 8
Proof :
8  5 = 3
8 ^ 2 + 5 ^ 2 = 64 + 25 = 89 
The product of two number is between 40 and 50. One number is 4 time less than the other