Question

the number the number X is 2 more than the number Y if the sum of the square of X and Y is 34 find the product of X and Y given x minus Y is equal to 2 and X square + Y square is equal to 34 find the value of x y​

Answer 1

$\underline{\huge\texttt{Answer:}}$

Given ,

$x - y = 2 \: \: \: \: \: \: \: .....(1)$

and ,

${x}^{2} + {y}^{2} = 34 \: \: \: \: \: \: .....(2)$

From , $(1)$ we have ,

$x - y = 2 \\ \\ \\ x = 2 + y \: \: \: \: \: \: \: .....(3)$

By putting the value of $x$ in equation$(2)$, we get —

${x}^{2} + {y}^{2} = 34 \\ \\ \\ = > {(2 + y)}^{2} + {y}^{2} = 34 \\ \\ \\ = > 4 + 4y + {y}^{2} + {y}^{2} = 34 \\ \\ \\ = > 2 {y}^{2} + 4y - 30 = 0 \: \: \: \: \: \: .....(4)$

By using the quadratic formula ,we have —

$y = \frac{ - b \binom{ + }{ - } \sqrt{ {b}^{2} - 4ac } }{2a} \\ \\ \\ \: \: \: = \frac{ - 4 \binom{ + }{ - } \sqrt{ {4}^{2} - 4.2.( - 30)} }{4} \\ \\ \\ = \frac{ - 4 \binom{ + }{ - } \sqrt{16 + 240} }{4} \\ \\ \\ = \frac{ - 4 \binom{ + }{ - } \sqrt{256} }{4 } \\ \\ \\ = \frac{ - 4 \binom{ + }{ - }16 }{4}$




Therefore ,




$y = \frac{ - 4 + 16}{4} \\ \\ \\ \: \: \: = 3$

And ,

$y = \frac{ - 4 - 16}{4} \\ \\ \\ \: \: \: = - 5$



Now , if y = 3 ,


By putting the value of y in equation (3) , we get :-

$x = 2 + 3 = 5$



and , if y = -5 ,


By putting the value of y in equation (3) , we get :-

$x = 2 + ( - 5) = - 3$


But , y = -3 won't satisfy , ${x}^{2} + {y}^{2} = 34$[ equation (2) ]


So , y = 3 ; and x = 5


Therefore , the product of the numbers

= 3 × 5

= 15 .