Question

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.

Answer 1

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

Answer 2

The product of two number is between 40 and 50. One number is 4 time less than the other