The difference of two positive numbers is 7 and the sum of their squares is 85. Find the numbers.?

You have :

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A - B = 7 ----------> B = A + 7

A^2 + B^2 = 85

A^2 + ( A + 7 )^2 = 85

A^2 + 49 + 14 A + A^2 = 85

2A^2 + 14A - 36 = 0

A = { (- 14 ) +/- SQRT [ ( 14 )^2 - ( 4 ) ( 2 ) ( -36 ) ] } / { ( 2 ) ( 2 ) }

A = { -14 +/- SQRT [ ( 196 ) + ( 288 ) ] } / { 4 }

A = { -14 + 22 } / { 4 } = 2

A = { --14 - 22 ) / { 4 } = - 9 --------> Unacceptable because negative

A = 2

B = A + 7 = 9

Answers: A = 2 B = 9 <------------------

Check the answers :

( A )^2 + ( B )^2 = ( 2 )^2 + ( 9 )^2 = 4 + 81 = 85 ---->[ OK ]

x = the greater number

x - 7 = the smaller number.

x ^2 + (x - 7 ) = 85

x^2 + x^2 - 14x + 49 = 85

2x^2 - 14x + 49 - 85 = 0

2x^2 - 14 x -36 = 0

Simplifying the equation

x^2 - 7x - 18 = 0

Factoring

(x - 9)(x +2 ) = 0

x - 9 = 0 ; x = 9 (valid solution) : The greater number

x + 2 = 0 ; x = -2

x - 7 = the smaller number

9 - 7 = 2 (the smaller number)

So, the numbers are 9 and 2.

2 and 9 :)

2 squared is 4

9 squared is 81

81 + 4 is 85