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# algebraic proof

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Apr 19, 2018

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Alright, here we go:

To prove that an expression is never positive, we just need to prove that the maximun value of the expression is smaller than zero.

We can just expand the expression.

\(3(x+1)(x+7)=3x^2+24x+21\)

\((2x+5)^2=4x^2+20x+25\)

\(3x^2+24x+21- (4x^2+20x+25) = -x^2+4x-4\)

Factoring out the negative one, we have:

\( -(x^2-4x+4)\)

Since

\((x^2-4x+4)\)

is a perfect square, we can rewrite the expression like this:

\(-(x-2)^2\)

Since a square is always positive, and a negative of a square is negative, we proved that the original expression is negative, and always will be negative.

I hope this answers your question and you have a wonderful day!

Apr 19, 2018
edited by GYanggg  Apr 19, 2018

#1
+982
+3

Alright, here we go:

To prove that an expression is never positive, we just need to prove that the maximun value of the expression is smaller than zero.

We can just expand the expression.

\(3(x+1)(x+7)=3x^2+24x+21\)

\((2x+5)^2=4x^2+20x+25\)

\(3x^2+24x+21- (4x^2+20x+25) = -x^2+4x-4\)

Factoring out the negative one, we have:

\( -(x^2-4x+4)\)

Since

\((x^2-4x+4)\)

is a perfect square, we can rewrite the expression like this:

\(-(x-2)^2\)

Since a square is always positive, and a negative of a square is negative, we proved that the original expression is negative, and always will be negative.

I hope this answers your question and you have a wonderful day!

GYanggg Apr 19, 2018
edited by GYanggg  Apr 19, 2018