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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
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ii + jj = 306
(24 - j)(24 - j) + jj = 306
576 - 48j + jj + jj = 306
2jj - 48j + 270 = 0
2j^2 - 48j + 270 = 0
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the above quadratic equation is in standard form, with a=2, b=-48, and c=270
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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
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the two real roots (i.e. the two solutions), of the quadratic are:
j = 15
j = 9
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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.
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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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