y + x = 30
xy = 209
y = 30 -x
x(30-x) = 209
30x - x^2 = 209
x^2 - 30x + 209 = 0
x = 19
y = 11
the difference is 19-11 = 8
The numbers are 11 and 19.
Let's assume that the first number be a and the second is b.
a + b = 30
ab = 209
Therefore, in equation 1,
[tex]a\times b = 209\\b= \dfrac{209}{a}[/tex]
substitute the value of b in equation 1,
[tex]a+b=30\\\\a+\dfrac{209}{a}=30\\\\\dfrac{a^2+209}{a} =30\\\\a^2 +209 = 30a\\a^2-30a +209 = 0[/tex]
[tex]a^2 -30a+209= 0\\a^2 -19a-11a+209= 0\\a(a-19)-11(a-19)=0\\(a-11)(a-19)=0[/tex]
Substituting the factor against 0,
[tex](a-11) = 0\\a = 11\\\\(a-19)=0\\a = 19[/tex]
Therefore, the numbers are 11 and 19.
Learn more about factorization:
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