Question

the sum of the two number is 11 and their product is 30 then the number

Answer 1

Let the two numbers be x and y.

Given that their sum = 11
Therefore
x + y = 11

Also given that their product is 30
Therefore
xy = 30
y = 30/x

x + y = 11
x + 30/x = 11 [Substituting 30/x in place of y]
x² + 30 = 11x
x² - 11x + 30 = 0
x² - 5x - 6x + 30 = 0
x(x-5) + 6(x-5) = 0
(x-5) (x+6) = 0
x = 5 or x = - 6

y = 30/x = 30/5 or 30/-6
y = 6 or - 5

If x = 5 then y = 6
If x = - 6 then y = - 5

Rejecting x = - 6 and y = -5 because the sum of these numbers is - 11 but given sum is 11.

Therefore the numbers are 5 and 6.

Answer 2

$\textbf{\underline{\underline{According\:to\:the\:Question}}}$

★Assume the two numbers be p and n

Then,

$\textbf{\underline{\underline{According\:to\:the\: Condition}}}$

p + n = 11 ...... (1)

p × n = 30 ...... (2)

★Here we have to solve equation (2)

$\Large{\fbox{Using\;Substitution\;method}}$

$\tt{\rightarrow n=\dfrac{30}{p}}$

★Putting the value of n in equation (1)

$\tt{\rightarrow p+\dfrac{30}{p}=11}$

p² - 11p + 30 = 0

$\Large{\fbox{Splitting\;the\;middle\;term}}$

p² - 6p - 5p + 30 = 0

p(p - 6) + 5(p - 6) = 0

(p - 6) = 0

p = 6

(p - 5) = 0

p = 5

★Here we get the numbers :-

$\Large{\fbox{Required\;number\;= 6\;and\;5}}$