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# the product of two numbers is six times their sum. the sum of their squares is 325. what are the two numbers?

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the product of two numbers is six times their sum. the sum of their squares is 325. what are the two numbers?

Guest Oct 13, 2014

#1
+17721
+5

I'm assuming that you want an exact answer, not a graphical approximation (which is a lost easier!):

x² + y²  =  325  ===>  x² + 2xy + y²  =  325 + 2xy  ===>  (x + y)²  =  325 + 2·6(x + y)

===>  (x + y)²  =  325 + 12(x + y)     ===>  (x + y)² - 12(x + y) - 325  =  0

Factor or use the quadratic formula:  [(x + y) - 25][(x + y ) + 13]  =  0

===>  x  =  25    or    x  =  -13

If x + y  =  25    ===>     y  =  25 - x

x² + y²  =  325   ===>   x² + (25 - x)²  =  325   ===>   x² + 625 - 50x + x²  =  325

===>   2x² - 50x + 300  =  0   ===>   x² - 25x + 150  =  0

Factoring:  (x - 15)(x - 10)  =  0   ===>  x  =  15     or     x  =  10

===>  when x  =  10,  y  =  15; when x = 15, y = 10

Now, x = -13  gives you two more answers; in a similar solution as the above:  x  =  (-13 ± √481 ) / 2

When x = (-13 + √481 ) / 2, y = (-13 - √481 ) / 2

and  when x = (-13 - √481 ) / 2, y = (-13 + √481 ) / 2

geno3141  Oct 13, 2014
Sort:

#1
+17721
+5

I'm assuming that you want an exact answer, not a graphical approximation (which is a lost easier!):

x² + y²  =  325  ===>  x² + 2xy + y²  =  325 + 2xy  ===>  (x + y)²  =  325 + 2·6(x + y)

===>  (x + y)²  =  325 + 12(x + y)     ===>  (x + y)² - 12(x + y) - 325  =  0

Factor or use the quadratic formula:  [(x + y) - 25][(x + y ) + 13]  =  0

===>  x  =  25    or    x  =  -13

If x + y  =  25    ===>     y  =  25 - x

x² + y²  =  325   ===>   x² + (25 - x)²  =  325   ===>   x² + 625 - 50x + x²  =  325

===>   2x² - 50x + 300  =  0   ===>   x² - 25x + 150  =  0

Factoring:  (x - 15)(x - 10)  =  0   ===>  x  =  15     or     x  =  10

===>  when x  =  10,  y  =  15; when x = 15, y = 10

Now, x = -13  gives you two more answers; in a similar solution as the above:  x  =  (-13 ± √481 ) / 2

When x = (-13 + √481 ) / 2, y = (-13 - √481 ) / 2

and  when x = (-13 - √481 ) / 2, y = (-13 + √481 ) / 2

geno3141  Oct 13, 2014
#2
+85817
0

Very crafty the way you did that one geno !!!!

CPhill  Oct 14, 2014

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