Question

3. The sum of two numbers is 18 and the sum of their squares is 194. Find the numbers.
(quadratic equation quation)​

Answer 1

Answer:

13,5

Step-by-step explanation:

let the two numbers be x and y

according to question

x+y= 18.....(1) and

x^2+y^2= 194......(2)

from equation 1..

x+y =18

》x= 18-y........(3)

substitute the value of x in equation 2

we get,

(18-y)^2 + y^2 = 194

》324 -36y+y^2+ y^2 =194

》2y^2-36y+(324-194) = 0

》y^2-18y+130=0

》y^2-13y-5y+65=0

》y(y-13)-5(y-13)=0

》(y-5)(y-13)=0

Gives us y=5,13

now putting value of y in equation 3

x = 13,5

THANK YOU...

HOPE IT HELPS

Answer 2

$\sf{\bold{\green{\underline{\underline{Given}}}}}$

• Sum of 2 numbers is 18
• Sum of squares of 2 numbers is 194

______________________

$\sf{\bold{\green{\underline{\underline{To\:Find}}}}}$

• Original numbers = ??

______________________

$\sf{\bold{\green{\underline{\underline{Solution}}}}}$

• let the numbers be x and y

⠀⠀⠀⠀

Acc. to 1st statement

⠀⠀⠀⠀

x + y = 18

⠀⠀

x = 18 - y ----- ( i )

⠀⠀⠀⠀

Acc. to 2nd statement :-

⠀⠀⠀⠀

${\bold{x^2 + y^2 = 194}}$

⠀⠀⠀⠀

• putting value of x from eq ( i )

⠀⠀⠀⠀

${\bold{( 18 - y ) ^2 + y^2 = 194}}$

⠀⠀⠀⠀

${\bold{324 + y^2 - 36y + y^2 = 194}}$

⠀⠀⠀⠀

${\bold{2y^2 - 36y + 324 = 194}}$

⠀⠀

${\bold{2y^2 - 36y + 324 - 194 = 0 }}$

⠀⠀⠀⠀

${\bold{2y^2 - 36y + 130 = 0}}$

⠀⠀⠀⠀

${\bold{2 ( y^2 - 18y + 65 ) = 0 }}$

⠀⠀⠀⠀

${\bold{y^2 - 18y + 65 = 0 / 2}}$

⠀⠀⠀⠀

${\bold{y^2 - 18y + 65 = 0 }}$

⠀⠀⠀⠀

${\bold{y^2 - 13y - 5y + 65 = 0 }}$

⠀⠀

${\bold{y ( y - 13 ) - 5 ( y - 13 ) = 0 }}$

⠀⠀⠀⠀

${\bold{( y - 13 ) ( y - 5 ) = 0 }}$

⠀⠀⠀⠀

$\begin{tabular}{|c|c|}\cline{1-2}\sf y - 13= 0 &\sf y - 5= 0 \\\cline{1-2}\sf y = 13 + 0 &\sf y = 0 + 5 \\\cline{1-2}\sf y = 13 &\sf y = 5\\\cline{1-2}\end{tabular}$

⠀⠀⠀⠀

• Putting value of y in eq ( i )

⠀⠀⠀⠀

$\begin{tabular}{|c|c|}\cline{1-2}\sf x + 5 = 18 &\sf x + 13 = 15 \\\cline{1-2}\sf x = 18 - 5 &\sf x = 18 - 13\\\cline{1-2}\sf x = 5 &\sf x = 13 \\\cline{1-2}\end{tabular}$

⠀⠀⠀⠀

______________________

$\sf{\bold{\green{\underline{\underline{Answer}}}}}$

• Therefore the two required numbers are 5 and 13