Question

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

Answer 1

`\left \{ {\big{x+y=34.5} \atop \big{xy=297}} \right.\\\\ \left \{ {\big{x=34.5-y\ \ \ \ \ } \atop\big {(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= (-69-3)/(2\cdot(-2)) = (-72)/(-4) =18\ \ \ \Rightarrow\ \ \ x_1=34.5-18=16.5\\\\y_2= (-69+3)/(2\cdot(-2)) = (-66)/(-4) =16.5\ \ \ \Rightarrow\ \ \ x_2=34.5-16.5=18`


`Ans.\ two\ numbers\ are:\ 18\ \ \ and\ \ \ 16.5`

Answer 2

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