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Prove that (4x-5)2 -5x(3x-8) is a positive for all values of x

User JRichardsz
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To prove that (4x - 5)^2 - 5x(3x - 8) is positive for all values of x, we can start by expanding the expression:

(4x - 5)^2 - 5x(3x - 8) = 16x^2 - 40x + 25 - 15x^2 + 40x = -14x^2 + 80x - 25

Now, we can complete the square to find that this expression is always positive:

-14x^2 + 80x - 25 = (-14x^2 + 80x) - 25

= (-14x^2 + 80x + (80/2)^2 - (80/2)^2) - 25

= (-14x + 40)^2 - 25 - 40^2 + 40^2

= 15^2 - 25

= (15 - 5)(15 + 5)

= 10(20)

= 200

Since this expression is equal to 200, which is a positive number, we can conclude that (4x - 5)^2 - 5x(3x - 8) is positive for all values of x.

User David Allan Finch
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