SOLUTION: The sum of two numbers is 24 . the sum of the squares of the two numbers is 306 . what is the product of the two numbers a). 119 b). 128 c). 135 d). 144

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Question 818659: The sum of two numbers is 24 . the sum of the squares of the two numbers is 306 . what is the product of the two numbers
a). 119
b). 128
c). 135
d). 144
Found 2 solutions by TimothyLamb, Edwin McCravy:
Answer by TimothyLamb(4379)   (Show Source): You can put this solution on YOUR website!
i + j = 24
i = (24 - j)
ii + jj = 306
---
ii + jj = 306
(24 - j)(24 - j) + jj = 306
576 - 48j + jj + jj = 306
2jj - 48j + 270 = 0
2j^2 - 48j + 270 = 0
---
the above quadratic equation is in standard form, with a=2, b=-48, and c=270
---
to solve the quadratic equation, by using the quadratic formula, plug this:
2 -48 270
into this: https://sooeet.com/math/quadratic-equation-solver.php
---
the two real roots (i.e. the two solutions), of the quadratic are:
j = 15
j = 9
---
answer:
the two numbers are 9 and 15
the product of the two numbers is 135
---
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Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!
The sum of two numbers is 24 . 
That says "x+y is 24" or x+y = 24

the sum of the squares of the two numbers is 306 . what is the product of the two numbers
That says "x²+y² is 306", or x²+y² = 306

what is the product of the two numbers
That says "What is xy?

So we have these two equations., 

(1)     x+y = 24
(2)   x²+y² = 306



        x+y = 24

Square both sides:

     (x+y)² = 24²

Multiply it out:

 (x+y)(x+y) = 576
  x²+2xy+y² = 576

Rearrange

(x²+y²)+2xy = 576

Use (2) about to substitute 306 for (x²+y²)

    306+2xy = 576

        2xy = 270

         xy = 135.

That's the answer.  

---------------------------------

It wasn't necessary to find x and y. We COULD find them. 
They are 9 and 15.  9+15=24 and 9²+15²=81+225=306,
and 9·15 = 135.

Edwin
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