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Show that  `25m^2-40mn+16n^2` is non-negative for all values of m and n.

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Show that  `25m^2-40mn+16n^2` is non-negative for all values of m and n.

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mjripalda's profile pic

Posted (Answer #1)

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To prove that `25m^2-40mn +16n^2` is non-negative for all values of m and n, set the polynomial greater than and equal to zero.

`25m^2-40mn +16n^2gt=0`

Then, factor.

`(5m-4n)(5m-4n)gt=0`

`(5m-4n)^2gt=0`

Then, consider this property of exponent which is:

`x^ngt=0 `   when n is an even number.

Since the exponent of the factor `5m-4n` is 2 which is an evennumber, then the inequality equation `(5m - 4n)^2gt=0` is always true.

Hence,  `25m^2-40mn +16n^2` is non-negative for any values of m and n.

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Posted (Answer #2)

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`25mn^2-40mn+16n^2=(5m-4n)^2>=0`

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