M^2+n^2=17 mn=3 (m-n)^2=?

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вопрос

Let's start by finding the values of m and n. We can solve the system of equations:

M^2 + n^2 = 17
Mn = 3

We can substitute the value of Mn into the first equation:

M^2 + (3/M)^2 = 17
M^2 + 9/M^2 = 17
M^4 - 17M^2 + 9 = 0

Now we can solve this quadratic equation by letting x = M^2:

x^2 - 17x + 9 = 0
(x - 9)(x - 1) = 0
x = 9 or x = 1

Since x = M^2, we have two possibilities for M:

M^2 = 9 or M^2 = 1
M = 3 or M = 1

Now we can find the corresponding values of n by substituting M back into the equation Mn = 3:

If M = 3, then n = 1
If M = 1, then n = 3

Now we can find (m-n)^2:

If M = 3 and n = 1:
(m-n)^2 = (3-1)^2 = 2^2 = 4

If M = 1 and n = 3:
(m-n)^2 = (1-3)^2 = (-2)^2 = 4

Therefore, (m-n)^2 = 4.