Find two real numbers whose sum is 34.5 and whose product is 297

X + y = 34.5
xy = 297
From the first one. y = 34.5 - x
Plug that into the second one. x(34.5 - x) = 297
Eliminate parentheses. 34.5x - x² = 297
Bend that around and tidy it up. x² - 34.5x + 297 = 0
Apply the quadratic formula to that, and find.
x =18
x = 16.5

\( \left \{ {x+y=34.5} \atop {xy=297} \right. \\\\ \left \{ {x=34.5-y\ \ \ \ \ } \atop {(34.5-y)y=297} \right. \\\\\\ 34.5y-y^2-297=0\ \ \ \Rightarrow\ \ \ -y^2+34.5y-297=0\ /\cdot 2\\\\-2y^2+69y-594=0\\\\ \ \ \ \Rightarrow\ \ \ \Delta=69^2-4\cdot(-2)\cdot(-594)=4761-4752=9\\\\y_1= \frac{-69-3}{2\cdot(-2)} = \frac{-72}{-4} =18\ \ \ \Rightarrow\ \ \ x_1=34.5-18=16.5\\\\y_2= \frac{-69+3}{2\cdot(-2)} = \frac{-66}{-4} =16.5\ \ \ \Rightarrow\ \ \ x_2=34.5-16.5=18 \)

\( Ans. \ two\ numbers\ are:\ 18\ \ \ and\ \ \ 16.5 \)


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