Prove that (4x–5)^2 – 5x(3x – 8) is positive for all values of x.
To begin we need to simplify the expression. First we multiply out (4x–5)^2 to get 16x2+40x+25 and then we multiply out 5x(3x – 8) to get 15x2-40x. This makes the whole expression 16x2+40x+25-(15x2-40x), which equals 16x2+40x+25-15x2+40x. This simplifies to x2+25. We know that x2 is positive for all values of x, and so x2+25 must also be positive for all values of x.
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